Calculus Examples

${(x + 1)}^{2} (2 x - x^{2})$

Step 1

Write ${(x + 1)}^{2} (2 x - x^{2})$ as a function.

$f (x) = {(x + 1)}^{2} (2 x - x^{2})$

Step 2

Find the second derivative.

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Step 2.1

Find the first derivative.

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Step 2.1.1

Rewrite ${(x + 1)}^{2}$ as $(x + 1) (x + 1)$ .

$\frac{d}{d x} [(x + 1) (x + 1) (2 x - x^{2})]$

Step 2.1.2

Expand $(x + 1) (x + 1)$ using the FOIL Method.

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Step 2.1.2.1

Apply the distributive property.

$\frac{d}{d x} [(x (x + 1) + 1 (x + 1)) (2 x - x^{2})]$

Step 2.1.2.2

Apply the distributive property.

$\frac{d}{d x} [(x \cdot x + x \cdot 1 + 1 (x + 1)) (2 x - x^{2})]$

Step 2.1.2.3

Apply the distributive property.

$\frac{d}{d x} [(x \cdot x + x \cdot 1 + 1 x + 1 \cdot 1) (2 x - x^{2})]$

Step 2.1.3

Simplify and combine like terms.

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Step 2.1.3.1

Simplify each term.

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Step 2.1.3.1.1

Multiply $x$ by $x$ .

$\frac{d}{d x} [(x^{2} + x \cdot 1 + 1 x + 1 \cdot 1) (2 x - x^{2})]$

Step 2.1.3.1.2

Multiply $x$ by $1$ .

$\frac{d}{d x} [(x^{2} + x + 1 x + 1 \cdot 1) (2 x - x^{2})]$

Step 2.1.3.1.3

Multiply $x$ by $1$ .

$\frac{d}{d x} [(x^{2} + x + x + 1 \cdot 1) (2 x - x^{2})]$

Step 2.1.3.1.4

Multiply $1$ by $1$ .

$\frac{d}{d x} [(x^{2} + x + x + 1) (2 x - x^{2})]$

Step 2.1.3.2

Add $x$ and $x$ .

$\frac{d}{d x} [(x^{2} + 2 x + 1) (2 x - x^{2})]$

Step 2.1.4

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x^{2} + 2 x + 1$ and $g (x) = 2 x - x^{2}$ .

$(x^{2} + 2 x + 1) \frac{d}{d x} [2 x - x^{2}] + (2 x - x^{2}) \frac{d}{d x} [x^{2} + 2 x + 1]$

Step 2.1.5

Differentiate.

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Step 2.1.5.1

By the Sum Rule, the derivative of $2 x - x^{2}$ with respect to $x$ is $\frac{d}{d x} [2 x] + \frac{d}{d x} [- x^{2}]$ .

$(x^{2} + 2 x + 1) (\frac{d}{d x} [2 x] + \frac{d}{d x} [- x^{2}]) + (2 x - x^{2}) \frac{d}{d x} [x^{2} + 2 x + 1]$

Step 2.1.5.2

Since $2$ is constant with respect to $x$ , the derivative of $2 x$ with respect to $x$ is $2 \frac{d}{d x} [x]$ .

$(x^{2} + 2 x + 1) (2 \frac{d}{d x} [x] + \frac{d}{d x} [- x^{2}]) + (2 x - x^{2}) \frac{d}{d x} [x^{2} + 2 x + 1]$

Step 2.1.5.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$(x^{2} + 2 x + 1) (2 \cdot 1 + \frac{d}{d x} [- x^{2}]) + (2 x - x^{2}) \frac{d}{d x} [x^{2} + 2 x + 1]$

Step 2.1.5.4

Multiply $2$ by $1$ .

$(x^{2} + 2 x + 1) (2 + \frac{d}{d x} [- x^{2}]) + (2 x - x^{2}) \frac{d}{d x} [x^{2} + 2 x + 1]$

Step 2.1.5.5

Since $- 1$ is constant with respect to $x$ , the derivative of $- x^{2}$ with respect to $x$ is $- \frac{d}{d x} [x^{2}]$ .

$(x^{2} + 2 x + 1) (2 - \frac{d}{d x} [x^{2}]) + (2 x - x^{2}) \frac{d}{d x} [x^{2} + 2 x + 1]$

Step 2.1.5.6

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$(x^{2} + 2 x + 1) (2 - (2 x)) + (2 x - x^{2}) \frac{d}{d x} [x^{2} + 2 x + 1]$

Step 2.1.5.7

Multiply $2$ by $- 1$ .

$(x^{2} + 2 x + 1) (2 - 2 x) + (2 x - x^{2}) \frac{d}{d x} [x^{2} + 2 x + 1]$

Step 2.1.5.8

By the Sum Rule, the derivative of $x^{2} + 2 x + 1$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [2 x] + \frac{d}{d x} [1]$ .

$(x^{2} + 2 x + 1) (2 - 2 x) + (2 x - x^{2}) (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [2 x] + \frac{d}{d x} [1])$

Step 2.1.5.9

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$(x^{2} + 2 x + 1) (2 - 2 x) + (2 x - x^{2}) (2 x + \frac{d}{d x} [2 x] + \frac{d}{d x} [1])$

Step 2.1.5.10

Since $2$ is constant with respect to $x$ , the derivative of $2 x$ with respect to $x$ is $2 \frac{d}{d x} [x]$ .

$(x^{2} + 2 x + 1) (2 - 2 x) + (2 x - x^{2}) (2 x + 2 \frac{d}{d x} [x] + \frac{d}{d x} [1])$

Step 2.1.5.11

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$(x^{2} + 2 x + 1) (2 - 2 x) + (2 x - x^{2}) (2 x + 2 \cdot 1 + \frac{d}{d x} [1])$

Step 2.1.5.12

Multiply $2$ by $1$ .

$(x^{2} + 2 x + 1) (2 - 2 x) + (2 x - x^{2}) (2 x + 2 + \frac{d}{d x} [1])$

Step 2.1.5.13

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$(x^{2} + 2 x + 1) (2 - 2 x) + (2 x - x^{2}) (2 x + 2 + 0)$

Step 2.1.5.14

Add $2 x + 2$ and $0$ .

$(x^{2} + 2 x + 1) (2 - 2 x) + (2 x - x^{2}) (2 x + 2)$

Step 2.1.6

Simplify.

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Step 2.1.6.1

Factor $1$ out of $(x^{2} + 2 x + 1) (2 - 2 x) + (2 x - x^{2}) (2 x + 2)$ .

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Step 2.1.6.1.1

Factor $1$ out of $(x^{2} + 2 x + 1) (2 - 2 x)$ .

$1 ((x^{2} + 2 x + 1) (2 - 2 x)) + (2 x - x^{2}) (2 x + 2)$

Step 2.1.6.1.2

Factor $1$ out of $(2 x - x^{2}) (2 x + 2)$ .

$1 ((x^{2} + 2 x + 1) (2 - 2 x)) + 1 ((2 x - x^{2}) (2 x + 2))$

Step 2.1.6.1.3

Factor $1$ out of $1 ((x^{2} + 2 x + 1) (2 - 2 x)) + 1 ((2 x - x^{2}) (2 x + 2))$ .

$1 ((x^{2} + 2 x + 1) (2 - 2 x) + (2 x - x^{2}) (2 x + 2))$

Step 2.1.6.2

Multiply $(x^{2} + 2 x + 1) (2 - 2 x) + (2 x - x^{2}) (2 x + 2)$ by $1$ .

$(x^{2} + 2 x + 1) (2 - 2 x) + (2 x - x^{2}) (2 x + 2)$

Step 2.1.6.3

Reorder terms.

$(2 - 2 x) (x^{2} + 2 x + 1) + (2 x + 2) (2 x - x^{2})$

Step 2.1.6.4

Simplify each term.

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Step 2.1.6.4.1

Expand $(2 - 2 x) (x^{2} + 2 x + 1)$ by multiplying each term in the first expression by each term in the second expression.

$2 x^{2} + 2 (2 x) + 2 \cdot 1 - 2 x \cdot x^{2} - 2 x (2 x) - 2 x \cdot 1 + (2 x + 2) (2 x - x^{2})$

Step 2.1.6.4.2

Simplify each term.

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Step 2.1.6.4.2.1

Multiply $2$ by $2$ .

$2 x^{2} + 4 x + 2 \cdot 1 - 2 x \cdot x^{2} - 2 x (2 x) - 2 x \cdot 1 + (2 x + 2) (2 x - x^{2})$

Step 2.1.6.4.2.2

Multiply $2$ by $1$ .

$2 x^{2} + 4 x + 2 - 2 x \cdot x^{2} - 2 x (2 x) - 2 x \cdot 1 + (2 x + 2) (2 x - x^{2})$

Step 2.1.6.4.2.3

Multiply $x$ by $x^{2}$ by adding the exponents.

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Step 2.1.6.4.2.3.1

Move $x^{2}$ .

$2 x^{2} + 4 x + 2 - 2 (x^{2} x) - 2 x (2 x) - 2 x \cdot 1 + (2 x + 2) (2 x - x^{2})$

Step 2.1.6.4.2.3.2

Multiply $x^{2}$ by $x$ .

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Step 2.1.6.4.2.3.2.1

Raise $x$ to the power of $1$ .

$2 x^{2} + 4 x + 2 - 2 (x^{2} x^{1}) - 2 x (2 x) - 2 x \cdot 1 + (2 x + 2) (2 x - x^{2})$

Step 2.1.6.4.2.3.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$2 x^{2} + 4 x + 2 - 2 x^{2 + 1} - 2 x (2 x) - 2 x \cdot 1 + (2 x + 2) (2 x - x^{2})$

Step 2.1.6.4.2.3.3

Add $2$ and $1$ .

$2 x^{2} + 4 x + 2 - 2 x^{3} - 2 x (2 x) - 2 x \cdot 1 + (2 x + 2) (2 x - x^{2})$

Step 2.1.6.4.2.4

Rewrite using the commutative property of multiplication.

$2 x^{2} + 4 x + 2 - 2 x^{3} - 2 \cdot 2 x \cdot x - 2 x \cdot 1 + (2 x + 2) (2 x - x^{2})$

Step 2.1.6.4.2.5

Multiply $x$ by $x$ by adding the exponents.

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Step 2.1.6.4.2.5.1

Move $x$ .

$2 x^{2} + 4 x + 2 - 2 x^{3} - 2 \cdot 2 (x \cdot x) - 2 x \cdot 1 + (2 x + 2) (2 x - x^{2})$

Step 2.1.6.4.2.5.2

Multiply $x$ by $x$ .

$2 x^{2} + 4 x + 2 - 2 x^{3} - 2 \cdot 2 x^{2} - 2 x \cdot 1 + (2 x + 2) (2 x - x^{2})$

Step 2.1.6.4.2.6

Multiply $- 2$ by $2$ .

$2 x^{2} + 4 x + 2 - 2 x^{3} - 4 x^{2} - 2 x \cdot 1 + (2 x + 2) (2 x - x^{2})$

Step 2.1.6.4.2.7

Multiply $- 2$ by $1$ .

$2 x^{2} + 4 x + 2 - 2 x^{3} - 4 x^{2} - 2 x + (2 x + 2) (2 x - x^{2})$

Step 2.1.6.4.3

Subtract $4 x^{2}$ from $2 x^{2}$ .

$- 2 x^{2} + 4 x + 2 - 2 x^{3} - 2 x + (2 x + 2) (2 x - x^{2})$

Step 2.1.6.4.4

Subtract $2 x$ from $4 x$ .

$- 2 x^{2} + 2 x + 2 - 2 x^{3} + (2 x + 2) (2 x - x^{2})$

Step 2.1.6.4.5

Expand $(2 x + 2) (2 x - x^{2})$ using the FOIL Method.

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Step 2.1.6.4.5.1

Apply the distributive property.

$- 2 x^{2} + 2 x + 2 - 2 x^{3} + 2 x (2 x - x^{2}) + 2 (2 x - x^{2})$

Step 2.1.6.4.5.2

Apply the distributive property.

$- 2 x^{2} + 2 x + 2 - 2 x^{3} + 2 x (2 x) + 2 x (- x^{2}) + 2 (2 x - x^{2})$

Step 2.1.6.4.5.3

Apply the distributive property.

$- 2 x^{2} + 2 x + 2 - 2 x^{3} + 2 x (2 x) + 2 x (- x^{2}) + 2 (2 x) + 2 (- x^{2})$

Step 2.1.6.4.6

Simplify and combine like terms.

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Step 2.1.6.4.6.1

Simplify each term.

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Step 2.1.6.4.6.1.1

Rewrite using the commutative property of multiplication.

$- 2 x^{2} + 2 x + 2 - 2 x^{3} + 2 \cdot 2 x \cdot x + 2 x (- x^{2}) + 2 (2 x) + 2 (- x^{2})$

Step 2.1.6.4.6.1.2

Multiply $x$ by $x$ by adding the exponents.

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Step 2.1.6.4.6.1.2.1

Move $x$ .

$- 2 x^{2} + 2 x + 2 - 2 x^{3} + 2 \cdot 2 (x \cdot x) + 2 x (- x^{2}) + 2 (2 x) + 2 (- x^{2})$

Step 2.1.6.4.6.1.2.2

Multiply $x$ by $x$ .

$- 2 x^{2} + 2 x + 2 - 2 x^{3} + 2 \cdot 2 x^{2} + 2 x (- x^{2}) + 2 (2 x) + 2 (- x^{2})$

Step 2.1.6.4.6.1.3

Multiply $2$ by $2$ .

$- 2 x^{2} + 2 x + 2 - 2 x^{3} + 4 x^{2} + 2 x (- x^{2}) + 2 (2 x) + 2 (- x^{2})$

Step 2.1.6.4.6.1.4

Rewrite using the commutative property of multiplication.

$- 2 x^{2} + 2 x + 2 - 2 x^{3} + 4 x^{2} + 2 \cdot - 1 x \cdot x^{2} + 2 (2 x) + 2 (- x^{2})$

Step 2.1.6.4.6.1.5

Multiply $x$ by $x^{2}$ by adding the exponents.

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Step 2.1.6.4.6.1.5.1

Move $x^{2}$ .

$- 2 x^{2} + 2 x + 2 - 2 x^{3} + 4 x^{2} + 2 \cdot - 1 (x^{2} x) + 2 (2 x) + 2 (- x^{2})$

Step 2.1.6.4.6.1.5.2

Multiply $x^{2}$ by $x$ .

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Step 2.1.6.4.6.1.5.2.1

Raise $x$ to the power of $1$ .

$- 2 x^{2} + 2 x + 2 - 2 x^{3} + 4 x^{2} + 2 \cdot - 1 (x^{2} x^{1}) + 2 (2 x) + 2 (- x^{2})$

Step 2.1.6.4.6.1.5.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$- 2 x^{2} + 2 x + 2 - 2 x^{3} + 4 x^{2} + 2 \cdot - 1 x^{2 + 1} + 2 (2 x) + 2 (- x^{2})$

Step 2.1.6.4.6.1.5.3

Add $2$ and $1$ .

$- 2 x^{2} + 2 x + 2 - 2 x^{3} + 4 x^{2} + 2 \cdot - 1 x^{3} + 2 (2 x) + 2 (- x^{2})$

Step 2.1.6.4.6.1.6

Multiply $2$ by $- 1$ .

$- 2 x^{2} + 2 x + 2 - 2 x^{3} + 4 x^{2} - 2 x^{3} + 2 (2 x) + 2 (- x^{2})$

Step 2.1.6.4.6.1.7

Multiply $2$ by $2$ .

$- 2 x^{2} + 2 x + 2 - 2 x^{3} + 4 x^{2} - 2 x^{3} + 4 x + 2 (- x^{2})$

Step 2.1.6.4.6.1.8

Multiply $- 1$ by $2$ .

$- 2 x^{2} + 2 x + 2 - 2 x^{3} + 4 x^{2} - 2 x^{3} + 4 x - 2 x^{2}$

Step 2.1.6.4.6.2

Subtract $2 x^{2}$ from $4 x^{2}$ .

$- 2 x^{2} + 2 x + 2 - 2 x^{3} + 2 x^{2} - 2 x^{3} + 4 x$

Step 2.1.6.5

Combine the opposite terms in $- 2 x^{2} + 2 x + 2 - 2 x^{3} + 2 x^{2} - 2 x^{3} + 4 x$ .

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Step 2.1.6.5.1

Add $- 2 x^{2}$ and $2 x^{2}$ .

$2 x + 2 - 2 x^{3} + 0 - 2 x^{3} + 4 x$

Step 2.1.6.5.2

Add $2 x + 2 - 2 x^{3}$ and $0$ .

$2 x + 2 - 2 x^{3} - 2 x^{3} + 4 x$

Step 2.1.6.6

Add $2 x$ and $4 x$ .

$2 - 2 x^{3} - 2 x^{3} + 6 x$

Step 2.1.6.7

Subtract $2 x^{3}$ from $- 2 x^{3}$ .

$f' (x) = 2 - 4 x^{3} + 6 x$

Step 2.2

Find the second derivative.

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Step 2.2.1

Differentiate.

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Step 2.2.1.1

By the Sum Rule, the derivative of $2 - 4 x^{3} + 6 x$ with respect to $x$ is $\frac{d}{d x} [2] + \frac{d}{d x} [- 4 x^{3}] + \frac{d}{d x} [6 x]$ .

$\frac{d}{d x} [2] + \frac{d}{d x} [- 4 x^{3}] + \frac{d}{d x} [6 x]$

Step 2.2.1.2

Since $2$ is constant with respect to $x$ , the derivative of $2$ with respect to $x$ is $0$ .

$0 + \frac{d}{d x} [- 4 x^{3}] + \frac{d}{d x} [6 x]$

Step 2.2.2

Evaluate $\frac{d}{d x} [- 4 x^{3}]$ .

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Step 2.2.2.1

Since $- 4$ is constant with respect to $x$ , the derivative of $- 4 x^{3}$ with respect to $x$ is $- 4 \frac{d}{d x} [x^{3}]$ .

$0 - 4 \frac{d}{d x} [x^{3}] + \frac{d}{d x} [6 x]$

Step 2.2.2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$0 - 4 (3 x^{2}) + \frac{d}{d x} [6 x]$

Step 2.2.2.3

Multiply $3$ by $- 4$ .

$0 - 12 x^{2} + \frac{d}{d x} [6 x]$

Step 2.2.3

Evaluate $\frac{d}{d x} [6 x]$ .

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Step 2.2.3.1

Since $6$ is constant with respect to $x$ , the derivative of $6 x$ with respect to $x$ is $6 \frac{d}{d x} [x]$ .

$0 - 12 x^{2} + 6 \frac{d}{d x} [x]$

Step 2.2.3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$0 - 12 x^{2} + 6 \cdot 1$

Step 2.2.3.3

Multiply $6$ by $1$ .

$0 - 12 x^{2} + 6$

Step 2.2.4

Subtract $12 x^{2}$ from $0$ .

$f'' (x) = - 12 x^{2} + 6$

Step 2.3

The second derivative of $f (x)$ with respect to $x$ is $- 12 x^{2} + 6$ .

$- 12 x^{2} + 6$

Step 3

Set the second derivative equal to $0$ then solve the equation $- 12 x^{2} + 6 = 0$ .

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Step 3.1

Set the second derivative equal to $0$ .

$- 12 x^{2} + 6 = 0$

Step 3.2

Subtract $6$ from both sides of the equation.

$- 12 x^{2} = - 6$

Step 3.3

Divide each term in $- 12 x^{2} = - 6$ by $- 12$ and simplify.

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Step 3.3.1

Divide each term in $- 12 x^{2} = - 6$ by $- 12$ .

$\frac{- 12 x^{2}}{- 12} = \frac{- 6}{- 12}$

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Cancel the common factor of $- 12$ .

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Step 3.3.2.1.1

Cancel the common factor.

$\frac{- 12 x^{2}}{- 12} = \frac{- 6}{- 12}$

Step 3.3.2.1.2

Divide $x^{2}$ by $1$ .

$x^{2} = \frac{- 6}{- 12}$

Step 3.3.3

Simplify the right side.

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Step 3.3.3.1

Cancel the common factor of $- 6$ and $- 12$ .

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Step 3.3.3.1.1

Factor $- 6$ out of $- 6$ .

$x^{2} = \frac{- 6 (1)}{- 12}$

Step 3.3.3.1.2

Cancel the common factors.

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Step 3.3.3.1.2.1

Factor $- 6$ out of $- 12$ .

$x^{2} = \frac{- 6 \cdot 1}{- 6 \cdot 2}$

Step 3.3.3.1.2.2

Cancel the common factor.

$x^{2} = \frac{- 6 \cdot 1}{- 6 \cdot 2}$

Step 3.3.3.1.2.3

Rewrite the expression.

$x^{2} = \frac{1}{2}$

Step 3.4

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{\frac{1}{2}}$

Step 3.5

Simplify $\pm \sqrt{\frac{1}{2}}$ .

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Step 3.5.1

Rewrite $\sqrt{\frac{1}{2}}$ as $\frac{\sqrt{1}}{\sqrt{2}}$ .

$x = \pm \frac{\sqrt{1}}{\sqrt{2}}$

Step 3.5.2

Any root of $1$ is $1$ .

$x = \pm \frac{1}{\sqrt{2}}$

Step 3.5.3

Multiply $\frac{1}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$x = \pm \frac{1}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}$

Step 3.5.4

Combine and simplify the denominator.

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Step 3.5.4.1

Multiply $\frac{1}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$x = \pm \frac{\sqrt{2}}{\sqrt{2} \sqrt{2}}$

Step 3.5.4.2

Raise $\sqrt{2}$ to the power of $1$ .

$x = \pm \frac{\sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}}$

Step 3.5.4.3

Raise $\sqrt{2}$ to the power of $1$ .

$x = \pm \frac{\sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}}$

Step 3.5.4.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x = \pm \frac{\sqrt{2}}{{\sqrt{2}}^{1 + 1}}$

Step 3.5.4.5

Add $1$ and $1$ .

$x = \pm \frac{\sqrt{2}}{{\sqrt{2}}^{2}}$

Step 3.5.4.6

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

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Step 3.5.4.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

$x = \pm \frac{\sqrt{2}}{{(2^{\frac{1}{2}})}^{2}}$

Step 3.5.4.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$x = \pm \frac{\sqrt{2}}{2^{\frac{1}{2} \cdot 2}}$

Step 3.5.4.6.3

Combine $\frac{1}{2}$ and $2$ .

$x = \pm \frac{\sqrt{2}}{2^{\frac{2}{2}}}$

Step 3.5.4.6.4

Cancel the common factor of $2$ .

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Step 3.5.4.6.4.1

Cancel the common factor.

$x = \pm \frac{\sqrt{2}}{2^{\frac{2}{2}}}$

Step 3.5.4.6.4.2

Rewrite the expression.

$x = \pm \frac{\sqrt{2}}{2^{1}}$

Step 3.5.4.6.5

Evaluate the exponent.

$x = \pm \frac{\sqrt{2}}{2}$

Step 3.6

The complete solution is the result of both the positive and negative portions of the solution.

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Step 3.6.1

First, use the positive value of the $\pm$ to find the first solution.

$x = \frac{\sqrt{2}}{2}$

Step 3.6.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - \frac{\sqrt{2}}{2}$

Step 3.6.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = \frac{\sqrt{2}}{2}, - \frac{\sqrt{2}}{2}$

Step 4

Find the points where the second derivative is $0$ .

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Step 4.1

Substitute $\frac{\sqrt{2}}{2}$ in $f (x) = {(x + 1)}^{2} (2 x - x^{2})$ to find the value of $y$ .

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Step 4.1.1

Replace the variable $x$ with $\frac{\sqrt{2}}{2}$ in the expression.

$f (\frac{\sqrt{2}}{2}) = {((\frac{\sqrt{2}}{2}) + 1)}^{2} (2 (\frac{\sqrt{2}}{2}) - {(\frac{\sqrt{2}}{2})}^{2})$

Step 4.1.2

Simplify the result.

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Step 4.1.2.1

Rewrite ${(\frac{\sqrt{2}}{2} + 1)}^{2}$ as $(\frac{\sqrt{2}}{2} + 1) (\frac{\sqrt{2}}{2} + 1)$ .

$f (\frac{\sqrt{2}}{2}) = (\frac{\sqrt{2}}{2} + 1) (\frac{\sqrt{2}}{2} + 1) (2 (\frac{\sqrt{2}}{2}) - {(\frac{\sqrt{2}}{2})}^{2})$

Step 4.1.2.2

Expand $(\frac{\sqrt{2}}{2} + 1) (\frac{\sqrt{2}}{2} + 1)$ using the FOIL Method.

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Step 4.1.2.2.1

Apply the distributive property.

$f (\frac{\sqrt{2}}{2}) = (\frac{\sqrt{2}}{2} \cdot (\frac{\sqrt{2}}{2} + 1) + 1 (\frac{\sqrt{2}}{2} + 1)) (2 (\frac{\sqrt{2}}{2}) - {(\frac{\sqrt{2}}{2})}^{2})$

Step 4.1.2.2.2

Apply the distributive property.

$f (\frac{\sqrt{2}}{2}) = (\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2} \cdot 1 + 1 (\frac{\sqrt{2}}{2} + 1)) (2 (\frac{\sqrt{2}}{2}) - {(\frac{\sqrt{2}}{2})}^{2})$

Step 4.1.2.2.3

Apply the distributive property.

$f (\frac{\sqrt{2}}{2}) = (\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2} \cdot 1 + 1 (\frac{\sqrt{2}}{2}) + 1 \cdot 1) (2 (\frac{\sqrt{2}}{2}) - {(\frac{\sqrt{2}}{2})}^{2})$

Step 4.1.2.3

Simplify and combine like terms.

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Step 4.1.2.3.1

Simplify each term.

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Step 4.1.2.3.1.1

Multiply $\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{2}}{2}$ .

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Step 4.1.2.3.1.1.1

Multiply $\frac{\sqrt{2}}{2}$ by $\frac{\sqrt{2}}{2}$ .

$f (\frac{\sqrt{2}}{2}) = (\frac{\sqrt{2} \sqrt{2}}{2 \cdot 2} + \frac{\sqrt{2}}{2} \cdot 1 + 1 (\frac{\sqrt{2}}{2}) + 1 \cdot 1) (2 (\frac{\sqrt{2}}{2}) - {(\frac{\sqrt{2}}{2})}^{2})$

Step 4.1.2.3.1.1.2

Raise $\sqrt{2}$ to the power of $1$ .

$f (\frac{\sqrt{2}}{2}) = (\frac{\sqrt{2} \sqrt{2}}{2 \cdot 2} + \frac{\sqrt{2}}{2} \cdot 1 + 1 (\frac{\sqrt{2}}{2}) + 1 \cdot 1) (2 (\frac{\sqrt{2}}{2}) - {(\frac{\sqrt{2}}{2})}^{2})$

Step 4.1.2.3.1.1.3

Raise $\sqrt{2}$ to the power of $1$ .

$f (\frac{\sqrt{2}}{2}) = (\frac{\sqrt{2} \sqrt{2}}{2 \cdot 2} + \frac{\sqrt{2}}{2} \cdot 1 + 1 (\frac{\sqrt{2}}{2}) + 1 \cdot 1) (2 (\frac{\sqrt{2}}{2}) - {(\frac{\sqrt{2}}{2})}^{2})$

Step 4.1.2.3.1.1.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f (\frac{\sqrt{2}}{2}) = (\frac{{\sqrt{2}}^{1 + 1}}{2 \cdot 2} + \frac{\sqrt{2}}{2} \cdot 1 + 1 (\frac{\sqrt{2}}{2}) + 1 \cdot 1) (2 (\frac{\sqrt{2}}{2}) - {(\frac{\sqrt{2}}{2})}^{2})$

Step 4.1.2.3.1.1.5

Add $1$ and $1$ .

$f (\frac{\sqrt{2}}{2}) = (\frac{{\sqrt{2}}^{2}}{2 \cdot 2} + \frac{\sqrt{2}}{2} \cdot 1 + 1 (\frac{\sqrt{2}}{2}) + 1 \cdot 1) (2 (\frac{\sqrt{2}}{2}) - {(\frac{\sqrt{2}}{2})}^{2})$

Step 4.1.2.3.1.1.6

Multiply $2$ by $2$ .

$f (\frac{\sqrt{2}}{2}) = (\frac{{\sqrt{2}}^{2}}{4} + \frac{\sqrt{2}}{2} \cdot 1 + 1 (\frac{\sqrt{2}}{2}) + 1 \cdot 1) (2 (\frac{\sqrt{2}}{2}) - {(\frac{\sqrt{2}}{2})}^{2})$

Step 4.1.2.3.1.2

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

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Step 4.1.2.3.1.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

$f (\frac{\sqrt{2}}{2}) = (\frac{{(2^{\frac{1}{2}})}^{2}}{4} + \frac{\sqrt{2}}{2} \cdot 1 + 1 (\frac{\sqrt{2}}{2}) + 1 \cdot 1) (2 (\frac{\sqrt{2}}{2}) - {(\frac{\sqrt{2}}{2})}^{2})$

Step 4.1.2.3.1.2.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$f (\frac{\sqrt{2}}{2}) = (\frac{2^{\frac{1}{2} \cdot 2}}{4} + \frac{\sqrt{2}}{2} \cdot 1 + 1 (\frac{\sqrt{2}}{2}) + 1 \cdot 1) (2 (\frac{\sqrt{2}}{2}) - {(\frac{\sqrt{2}}{2})}^{2})$

Step 4.1.2.3.1.2.3

Combine $\frac{1}{2}$ and $2$ .

$f (\frac{\sqrt{2}}{2}) = (\frac{2^{\frac{2}{2}}}{4} + \frac{\sqrt{2}}{2} \cdot 1 + 1 (\frac{\sqrt{2}}{2}) + 1 \cdot 1) (2 (\frac{\sqrt{2}}{2}) - {(\frac{\sqrt{2}}{2})}^{2})$

Step 4.1.2.3.1.2.4

Cancel the common factor of $2$ .

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Step 4.1.2.3.1.2.4.1

Cancel the common factor.

$f (\frac{\sqrt{2}}{2}) = (\frac{2^{\frac{2}{2}}}{4} + \frac{\sqrt{2}}{2} \cdot 1 + 1 (\frac{\sqrt{2}}{2}) + 1 \cdot 1) (2 (\frac{\sqrt{2}}{2}) - {(\frac{\sqrt{2}}{2})}^{2})$

Step 4.1.2.3.1.2.4.2

Rewrite the expression.

$f (\frac{\sqrt{2}}{2}) = (\frac{2}{4} + \frac{\sqrt{2}}{2} \cdot 1 + 1 (\frac{\sqrt{2}}{2}) + 1 \cdot 1) (2 (\frac{\sqrt{2}}{2}) - {(\frac{\sqrt{2}}{2})}^{2})$

Step 4.1.2.3.1.2.5

Evaluate the exponent.

$f (\frac{\sqrt{2}}{2}) = (\frac{2}{4} + \frac{\sqrt{2}}{2} \cdot 1 + 1 (\frac{\sqrt{2}}{2}) + 1 \cdot 1) (2 (\frac{\sqrt{2}}{2}) - {(\frac{\sqrt{2}}{2})}^{2})$

Step 4.1.2.3.1.3

Cancel the common factor of $2$ and $4$ .

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Step 4.1.2.3.1.3.1

Factor $2$ out of $2$ .

$f (\frac{\sqrt{2}}{2}) = (\frac{2 (1)}{4} + \frac{\sqrt{2}}{2} \cdot 1 + 1 (\frac{\sqrt{2}}{2}) + 1 \cdot 1) (2 (\frac{\sqrt{2}}{2}) - {(\frac{\sqrt{2}}{2})}^{2})$

Step 4.1.2.3.1.3.2

Cancel the common factors.

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Step 4.1.2.3.1.3.2.1

Factor $2$ out of $4$ .

$f (\frac{\sqrt{2}}{2}) = (\frac{2 \cdot 1}{2 \cdot 2} + \frac{\sqrt{2}}{2} \cdot 1 + 1 (\frac{\sqrt{2}}{2}) + 1 \cdot 1) (2 (\frac{\sqrt{2}}{2}) - {(\frac{\sqrt{2}}{2})}^{2})$

Step 4.1.2.3.1.3.2.2

Cancel the common factor.

$f (\frac{\sqrt{2}}{2}) = (\frac{2 \cdot 1}{2 \cdot 2} + \frac{\sqrt{2}}{2} \cdot 1 + 1 (\frac{\sqrt{2}}{2}) + 1 \cdot 1) (2 (\frac{\sqrt{2}}{2}) - {(\frac{\sqrt{2}}{2})}^{2})$

Step 4.1.2.3.1.3.2.3

Rewrite the expression.

$f (\frac{\sqrt{2}}{2}) = (\frac{1}{2} + \frac{\sqrt{2}}{2} \cdot 1 + 1 (\frac{\sqrt{2}}{2}) + 1 \cdot 1) (2 (\frac{\sqrt{2}}{2}) - {(\frac{\sqrt{2}}{2})}^{2})$

Step 4.1.2.3.1.4

Multiply $\frac{\sqrt{2}}{2}$ by $1$ .

$f (\frac{\sqrt{2}}{2}) = (\frac{1}{2} + \frac{\sqrt{2}}{2} + 1 (\frac{\sqrt{2}}{2}) + 1 \cdot 1) (2 (\frac{\sqrt{2}}{2}) - {(\frac{\sqrt{2}}{2})}^{2})$

Step 4.1.2.3.1.5

Multiply $\frac{\sqrt{2}}{2}$ by $1$ .

$f (\frac{\sqrt{2}}{2}) = (\frac{1}{2} + \frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2} + 1 \cdot 1) (2 (\frac{\sqrt{2}}{2}) - {(\frac{\sqrt{2}}{2})}^{2})$

Step 4.1.2.3.1.6

Multiply $1$ by $1$ .

$f (\frac{\sqrt{2}}{2}) = (\frac{1}{2} + \frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2} + 1) (2 (\frac{\sqrt{2}}{2}) - {(\frac{\sqrt{2}}{2})}^{2})$

Step 4.1.2.3.2

Write $1$ as a fraction with a common denominator.

$f (\frac{\sqrt{2}}{2}) = (\frac{1}{2} + \frac{2}{2} + \frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2}) (2 (\frac{\sqrt{2}}{2}) - {(\frac{\sqrt{2}}{2})}^{2})$

Step 4.1.2.3.3

Combine the numerators over the common denominator.

$f (\frac{\sqrt{2}}{2}) = (\frac{1 + 2}{2} + \frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2}) (2 (\frac{\sqrt{2}}{2}) - {(\frac{\sqrt{2}}{2})}^{2})$

Step 4.1.2.3.4

Add $1$ and $2$ .

$f (\frac{\sqrt{2}}{2}) = (\frac{3}{2} + \frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2}) (2 (\frac{\sqrt{2}}{2}) - {(\frac{\sqrt{2}}{2})}^{2})$

Step 4.1.2.3.5

Combine the numerators over the common denominator.

$f (\frac{\sqrt{2}}{2}) = (\frac{3}{2} + \frac{\sqrt{2} + \sqrt{2}}{2}) (2 (\frac{\sqrt{2}}{2}) - {(\frac{\sqrt{2}}{2})}^{2})$

Step 4.1.2.4

Simplify terms.

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Step 4.1.2.4.1

Combine the numerators over the common denominator.

$f (\frac{\sqrt{2}}{2}) = \frac{3 + \sqrt{2} + \sqrt{2}}{2} \cdot (2 (\frac{\sqrt{2}}{2}) - {(\frac{\sqrt{2}}{2})}^{2})$

Step 4.1.2.4.2

Add $\sqrt{2}$ and $\sqrt{2}$ .

$f (\frac{\sqrt{2}}{2}) = \frac{3 + 2 \sqrt{2}}{2} \cdot (2 (\frac{\sqrt{2}}{2}) - {(\frac{\sqrt{2}}{2})}^{2})$

Step 4.1.2.5

Simplify each term.

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Step 4.1.2.5.1

Cancel the common factor of $2$ .

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Step 4.1.2.5.1.1

Cancel the common factor.

$f (\frac{\sqrt{2}}{2}) = \frac{3 + 2 \sqrt{2}}{2} \cdot (2 (\frac{\sqrt{2}}{2}) - {(\frac{\sqrt{2}}{2})}^{2})$

Step 4.1.2.5.1.2

Rewrite the expression.

$f (\frac{\sqrt{2}}{2}) = \frac{3 + 2 \sqrt{2}}{2} \cdot (\sqrt{2} - {(\frac{\sqrt{2}}{2})}^{2})$

Step 4.1.2.5.2

Apply the product rule to $\frac{\sqrt{2}}{2}$ .

$f (\frac{\sqrt{2}}{2}) = \frac{3 + 2 \sqrt{2}}{2} \cdot (\sqrt{2} - \frac{{\sqrt{2}}^{2}}{2^{2}})$

Step 4.1.2.5.3

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

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Step 4.1.2.5.3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

$f (\frac{\sqrt{2}}{2}) = \frac{3 + 2 \sqrt{2}}{2} \cdot (\sqrt{2} - \frac{{(2^{\frac{1}{2}})}^{2}}{2^{2}})$

Step 4.1.2.5.3.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$f (\frac{\sqrt{2}}{2}) = \frac{3 + 2 \sqrt{2}}{2} \cdot (\sqrt{2} - \frac{2^{\frac{1}{2} \cdot 2}}{2^{2}})$

Step 4.1.2.5.3.3

Combine $\frac{1}{2}$ and $2$ .

$f (\frac{\sqrt{2}}{2}) = \frac{3 + 2 \sqrt{2}}{2} \cdot (\sqrt{2} - \frac{2^{\frac{2}{2}}}{2^{2}})$

Step 4.1.2.5.3.4

Cancel the common factor of $2$ .

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Step 4.1.2.5.3.4.1

Cancel the common factor.

$f (\frac{\sqrt{2}}{2}) = \frac{3 + 2 \sqrt{2}}{2} \cdot (\sqrt{2} - \frac{2^{\frac{2}{2}}}{2^{2}})$

Step 4.1.2.5.3.4.2

Rewrite the expression.

$f (\frac{\sqrt{2}}{2}) = \frac{3 + 2 \sqrt{2}}{2} \cdot (\sqrt{2} - \frac{2}{2^{2}})$

Step 4.1.2.5.3.5

Evaluate the exponent.

$f (\frac{\sqrt{2}}{2}) = \frac{3 + 2 \sqrt{2}}{2} \cdot (\sqrt{2} - \frac{2}{2^{2}})$

Step 4.1.2.5.4

Raise $2$ to the power of $2$ .

$f (\frac{\sqrt{2}}{2}) = \frac{3 + 2 \sqrt{2}}{2} \cdot (\sqrt{2} - \frac{2}{4})$

Step 4.1.2.5.5

Cancel the common factor of $2$ and $4$ .

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Step 4.1.2.5.5.1

Factor $2$ out of $2$ .

$f (\frac{\sqrt{2}}{2}) = \frac{3 + 2 \sqrt{2}}{2} \cdot (\sqrt{2} - \frac{2 (1)}{4})$

Step 4.1.2.5.5.2

Cancel the common factors.

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Step 4.1.2.5.5.2.1

Factor $2$ out of $4$ .

$f (\frac{\sqrt{2}}{2}) = \frac{3 + 2 \sqrt{2}}{2} \cdot (\sqrt{2} - \frac{2 \cdot 1}{2 \cdot 2})$

Step 4.1.2.5.5.2.2

Cancel the common factor.

$f (\frac{\sqrt{2}}{2}) = \frac{3 + 2 \sqrt{2}}{2} \cdot (\sqrt{2} - \frac{2 \cdot 1}{2 \cdot 2})$

Step 4.1.2.5.5.2.3

Rewrite the expression.

$f (\frac{\sqrt{2}}{2}) = \frac{3 + 2 \sqrt{2}}{2} \cdot (\sqrt{2} - \frac{1}{2})$

Step 4.1.2.6

Simplify terms.

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Step 4.1.2.6.1

Apply the distributive property.

$f (\frac{\sqrt{2}}{2}) = \frac{3 + 2 \sqrt{2}}{2} \cdot \sqrt{2} + \frac{3 + 2 \sqrt{2}}{2} \cdot (- \frac{1}{2})$

Step 4.1.2.6.2

Combine $\frac{3 + 2 \sqrt{2}}{2}$ and $\sqrt{2}$ .

$f (\frac{\sqrt{2}}{2}) = \frac{(3 + 2 \sqrt{2}) \sqrt{2}}{2} + \frac{3 + 2 \sqrt{2}}{2} \cdot (- \frac{1}{2})$

Step 4.1.2.7

Multiply $\frac{3 + 2 \sqrt{2}}{2} (- \frac{1}{2})$ .

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Step 4.1.2.7.1

Multiply $\frac{3 + 2 \sqrt{2}}{2}$ by $\frac{1}{2}$ .

$f (\frac{\sqrt{2}}{2}) = \frac{(3 + 2 \sqrt{2}) \sqrt{2}}{2} - \frac{3 + 2 \sqrt{2}}{2 \cdot 2}$

Step 4.1.2.7.2

Multiply $2$ by $2$ .

$f (\frac{\sqrt{2}}{2}) = \frac{(3 + 2 \sqrt{2}) \sqrt{2}}{2} - \frac{3 + 2 \sqrt{2}}{4}$

Step 4.1.2.8

Simplify each term.

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Step 4.1.2.8.1

Apply the distributive property.

$f (\frac{\sqrt{2}}{2}) = \frac{3 \sqrt{2} + 2 \sqrt{2} \sqrt{2}}{2} - \frac{3 + 2 \sqrt{2}}{4}$

Step 4.1.2.8.2

Multiply $2 \sqrt{2} \sqrt{2}$ .

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Step 4.1.2.8.2.1

Raise $\sqrt{2}$ to the power of $1$ .

$f (\frac{\sqrt{2}}{2}) = \frac{3 \sqrt{2} + 2 (\sqrt{2} \sqrt{2})}{2} - \frac{3 + 2 \sqrt{2}}{4}$

Step 4.1.2.8.2.2

Raise $\sqrt{2}$ to the power of $1$ .

$f (\frac{\sqrt{2}}{2}) = \frac{3 \sqrt{2} + 2 (\sqrt{2} \sqrt{2})}{2} - \frac{3 + 2 \sqrt{2}}{4}$

Step 4.1.2.8.2.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f (\frac{\sqrt{2}}{2}) = \frac{3 \sqrt{2} + 2 {\sqrt{2}}^{1 + 1}}{2} - \frac{3 + 2 \sqrt{2}}{4}$

Step 4.1.2.8.2.4

Add $1$ and $1$ .

$f (\frac{\sqrt{2}}{2}) = \frac{3 \sqrt{2} + 2 {\sqrt{2}}^{2}}{2} - \frac{3 + 2 \sqrt{2}}{4}$

Step 4.1.2.8.3

Simplify each term.

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Step 4.1.2.8.3.1

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

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Step 4.1.2.8.3.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

$f (\frac{\sqrt{2}}{2}) = \frac{3 \sqrt{2} + 2 {(2^{\frac{1}{2}})}^{2}}{2} - \frac{3 + 2 \sqrt{2}}{4}$

Step 4.1.2.8.3.1.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$f (\frac{\sqrt{2}}{2}) = \frac{3 \sqrt{2} + 2 \cdot 2^{\frac{1}{2} \cdot 2}}{2} - \frac{3 + 2 \sqrt{2}}{4}$

Step 4.1.2.8.3.1.3

Combine $\frac{1}{2}$ and $2$ .

$f (\frac{\sqrt{2}}{2}) = \frac{3 \sqrt{2} + 2 \cdot 2^{\frac{2}{2}}}{2} - \frac{3 + 2 \sqrt{2}}{4}$

Step 4.1.2.8.3.1.4

Cancel the common factor of $2$ .

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Step 4.1.2.8.3.1.4.1

Cancel the common factor.

$f (\frac{\sqrt{2}}{2}) = \frac{3 \sqrt{2} + 2 \cdot 2^{\frac{2}{2}}}{2} - \frac{3 + 2 \sqrt{2}}{4}$

Step 4.1.2.8.3.1.4.2

Rewrite the expression.

$f (\frac{\sqrt{2}}{2}) = \frac{3 \sqrt{2} + 2 \cdot 2}{2} - \frac{3 + 2 \sqrt{2}}{4}$

Step 4.1.2.8.3.1.5

Evaluate the exponent.

$f (\frac{\sqrt{2}}{2}) = \frac{3 \sqrt{2} + 2 \cdot 2}{2} - \frac{3 + 2 \sqrt{2}}{4}$

Step 4.1.2.8.3.2

Multiply $2$ by $2$ .

$f (\frac{\sqrt{2}}{2}) = \frac{3 \sqrt{2} + 4}{2} - \frac{3 + 2 \sqrt{2}}{4}$

Step 4.1.2.9

To write $\frac{3 \sqrt{2} + 4}{2}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$f (\frac{\sqrt{2}}{2}) = \frac{3 \sqrt{2} + 4}{2} \cdot \frac{2}{2} - \frac{3 + 2 \sqrt{2}}{4}$

Step 4.1.2.10

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1.2.10.1

Multiply $\frac{3 \sqrt{2} + 4}{2}$ by $\frac{2}{2}$ .

$f (\frac{\sqrt{2}}{2}) = \frac{(3 \sqrt{2} + 4) \cdot 2}{2 \cdot 2} - \frac{3 + 2 \sqrt{2}}{4}$

Step 4.1.2.10.2

Multiply $2$ by $2$ .

$f (\frac{\sqrt{2}}{2}) = \frac{(3 \sqrt{2} + 4) \cdot 2}{4} - \frac{3 + 2 \sqrt{2}}{4}$

Step 4.1.2.11

Combine the numerators over the common denominator.

$f (\frac{\sqrt{2}}{2}) = \frac{(3 \sqrt{2} + 4) \cdot 2 - (3 + 2 \sqrt{2})}{4}$

Step 4.1.2.12

Simplify the numerator.

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Step 4.1.2.12.1

Apply the distributive property.

$f (\frac{\sqrt{2}}{2}) = \frac{3 \sqrt{2} \cdot 2 + 4 \cdot 2 - (3 + 2 \sqrt{2})}{4}$

Step 4.1.2.12.2

Multiply $2$ by $3$ .

$f (\frac{\sqrt{2}}{2}) = \frac{6 \sqrt{2} + 4 \cdot 2 - (3 + 2 \sqrt{2})}{4}$

Step 4.1.2.12.3

Multiply $4$ by $2$ .

$f (\frac{\sqrt{2}}{2}) = \frac{6 \sqrt{2} + 8 - (3 + 2 \sqrt{2})}{4}$

Step 4.1.2.12.4

Apply the distributive property.

$f (\frac{\sqrt{2}}{2}) = \frac{6 \sqrt{2} + 8 - 1 \cdot 3 - (2 \sqrt{2})}{4}$

Step 4.1.2.12.5

Multiply $- 1$ by $3$ .

$f (\frac{\sqrt{2}}{2}) = \frac{6 \sqrt{2} + 8 - 3 - (2 \sqrt{2})}{4}$

Step 4.1.2.12.6

Multiply $2$ by $- 1$ .

$f (\frac{\sqrt{2}}{2}) = \frac{6 \sqrt{2} + 8 - 3 - 2 \sqrt{2}}{4}$

Step 4.1.2.12.7

Subtract $2 \sqrt{2}$ from $6 \sqrt{2}$ .

$f (\frac{\sqrt{2}}{2}) = \frac{8 - 3 + 4 \sqrt{2}}{4}$

Step 4.1.2.12.8

Subtract $3$ from $8$ .

$f (\frac{\sqrt{2}}{2}) = \frac{5 + 4 \sqrt{2}}{4}$

Step 4.1.2.13

The final answer is $\frac{5 + 4 \sqrt{2}}{4}$ .

$\frac{5 + 4 \sqrt{2}}{4}$

Step 4.2

The point found by substituting $\frac{\sqrt{2}}{2}$ in $f (x) = {(x + 1)}^{2} (2 x - x^{2})$ is $(\frac{\sqrt{2}}{2}, \frac{5 + 4 \sqrt{2}}{4})$ . This point can be an inflection point.

$(\frac{\sqrt{2}}{2}, \frac{5 + 4 \sqrt{2}}{4})$

Step 4.3

Substitute $- \frac{\sqrt{2}}{2}$ in $f (x) = {(x + 1)}^{2} (2 x - x^{2})$ to find the value of $y$ .

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Step 4.3.1

Replace the variable $x$ with $- \frac{\sqrt{2}}{2}$ in the expression.

$f (- \frac{\sqrt{2}}{2}) = {((- \frac{\sqrt{2}}{2}) + 1)}^{2} (2 (- \frac{\sqrt{2}}{2}) - {(- \frac{\sqrt{2}}{2})}^{2})$

Step 4.3.2

Simplify the result.

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Step 4.3.2.1

Rewrite ${(- \frac{\sqrt{2}}{2} + 1)}^{2}$ as $(- \frac{\sqrt{2}}{2} + 1) (- \frac{\sqrt{2}}{2} + 1)$ .

$f (- \frac{\sqrt{2}}{2}) = (- \frac{\sqrt{2}}{2} + 1) (- \frac{\sqrt{2}}{2} + 1) (2 (- \frac{\sqrt{2}}{2}) - {(- \frac{\sqrt{2}}{2})}^{2})$

Step 4.3.2.2

Expand $(- \frac{\sqrt{2}}{2} + 1) (- \frac{\sqrt{2}}{2} + 1)$ using the FOIL Method.

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Step 4.3.2.2.1

Apply the distributive property.

$f (- \frac{\sqrt{2}}{2}) = (- \frac{\sqrt{2}}{2} \cdot (- \frac{\sqrt{2}}{2} + 1) + 1 (- \frac{\sqrt{2}}{2} + 1)) (2 (- \frac{\sqrt{2}}{2}) - {(- \frac{\sqrt{2}}{2})}^{2})$

Step 4.3.2.2.2

Apply the distributive property.

$f (- \frac{\sqrt{2}}{2}) = (- \frac{\sqrt{2}}{2} \cdot (- \frac{\sqrt{2}}{2}) - \frac{\sqrt{2}}{2} \cdot 1 + 1 (- \frac{\sqrt{2}}{2} + 1)) (2 (- \frac{\sqrt{2}}{2}) - {(- \frac{\sqrt{2}}{2})}^{2})$

Step 4.3.2.2.3

Apply the distributive property.

$f (- \frac{\sqrt{2}}{2}) = (- \frac{\sqrt{2}}{2} \cdot (- \frac{\sqrt{2}}{2}) - \frac{\sqrt{2}}{2} \cdot 1 + 1 (- \frac{\sqrt{2}}{2}) + 1 \cdot 1) (2 (- \frac{\sqrt{2}}{2}) - {(- \frac{\sqrt{2}}{2})}^{2})$

Step 4.3.2.3

Simplify and combine like terms.

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Step 4.3.2.3.1

Simplify each term.

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Step 4.3.2.3.1.1

Multiply $- \frac{\sqrt{2}}{2} (- \frac{\sqrt{2}}{2})$ .

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Step 4.3.2.3.1.1.1

Multiply $- 1$ by $- 1$ .

$f (- \frac{\sqrt{2}}{2}) = (1 (\frac{\sqrt{2}}{2}) \cdot \frac{\sqrt{2}}{2} - \frac{\sqrt{2}}{2} \cdot 1 + 1 (- \frac{\sqrt{2}}{2}) + 1 \cdot 1) (2 (- \frac{\sqrt{2}}{2}) - {(- \frac{\sqrt{2}}{2})}^{2})$

Step 4.3.2.3.1.1.2

Multiply $\frac{\sqrt{2}}{2}$ by $1$ .

$f (- \frac{\sqrt{2}}{2}) = (\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{2}}{2} - \frac{\sqrt{2}}{2} \cdot 1 + 1 (- \frac{\sqrt{2}}{2}) + 1 \cdot 1) (2 (- \frac{\sqrt{2}}{2}) - {(- \frac{\sqrt{2}}{2})}^{2})$

Step 4.3.2.3.1.1.3

Multiply $\frac{\sqrt{2}}{2}$ by $\frac{\sqrt{2}}{2}$ .

$f (- \frac{\sqrt{2}}{2}) = (\frac{\sqrt{2} \sqrt{2}}{2 \cdot 2} - \frac{\sqrt{2}}{2} \cdot 1 + 1 (- \frac{\sqrt{2}}{2}) + 1 \cdot 1) (2 (- \frac{\sqrt{2}}{2}) - {(- \frac{\sqrt{2}}{2})}^{2})$

Step 4.3.2.3.1.1.4

Raise $\sqrt{2}$ to the power of $1$ .

$f (- \frac{\sqrt{2}}{2}) = (\frac{\sqrt{2} \sqrt{2}}{2 \cdot 2} - \frac{\sqrt{2}}{2} \cdot 1 + 1 (- \frac{\sqrt{2}}{2}) + 1 \cdot 1) (2 (- \frac{\sqrt{2}}{2}) - {(- \frac{\sqrt{2}}{2})}^{2})$

Step 4.3.2.3.1.1.5

Raise $\sqrt{2}$ to the power of $1$ .

$f (- \frac{\sqrt{2}}{2}) = (\frac{\sqrt{2} \sqrt{2}}{2 \cdot 2} - \frac{\sqrt{2}}{2} \cdot 1 + 1 (- \frac{\sqrt{2}}{2}) + 1 \cdot 1) (2 (- \frac{\sqrt{2}}{2}) - {(- \frac{\sqrt{2}}{2})}^{2})$

Step 4.3.2.3.1.1.6

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f (- \frac{\sqrt{2}}{2}) = (\frac{{\sqrt{2}}^{1 + 1}}{2 \cdot 2} - \frac{\sqrt{2}}{2} \cdot 1 + 1 (- \frac{\sqrt{2}}{2}) + 1 \cdot 1) (2 (- \frac{\sqrt{2}}{2}) - {(- \frac{\sqrt{2}}{2})}^{2})$

Step 4.3.2.3.1.1.7

Add $1$ and $1$ .

$f (- \frac{\sqrt{2}}{2}) = (\frac{{\sqrt{2}}^{2}}{2 \cdot 2} - \frac{\sqrt{2}}{2} \cdot 1 + 1 (- \frac{\sqrt{2}}{2}) + 1 \cdot 1) (2 (- \frac{\sqrt{2}}{2}) - {(- \frac{\sqrt{2}}{2})}^{2})$

Step 4.3.2.3.1.1.8

Multiply $2$ by $2$ .

$f (- \frac{\sqrt{2}}{2}) = (\frac{{\sqrt{2}}^{2}}{4} - \frac{\sqrt{2}}{2} \cdot 1 + 1 (- \frac{\sqrt{2}}{2}) + 1 \cdot 1) (2 (- \frac{\sqrt{2}}{2}) - {(- \frac{\sqrt{2}}{2})}^{2})$

Step 4.3.2.3.1.2

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

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Step 4.3.2.3.1.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

$f (- \frac{\sqrt{2}}{2}) = (\frac{{(2^{\frac{1}{2}})}^{2}}{4} - \frac{\sqrt{2}}{2} \cdot 1 + 1 (- \frac{\sqrt{2}}{2}) + 1 \cdot 1) (2 (- \frac{\sqrt{2}}{2}) - {(- \frac{\sqrt{2}}{2})}^{2})$

Step 4.3.2.3.1.2.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$f (- \frac{\sqrt{2}}{2}) = (\frac{2^{\frac{1}{2} \cdot 2}}{4} - \frac{\sqrt{2}}{2} \cdot 1 + 1 (- \frac{\sqrt{2}}{2}) + 1 \cdot 1) (2 (- \frac{\sqrt{2}}{2}) - {(- \frac{\sqrt{2}}{2})}^{2})$

Step 4.3.2.3.1.2.3

Combine $\frac{1}{2}$ and $2$ .

$f (- \frac{\sqrt{2}}{2}) = (\frac{2^{\frac{2}{2}}}{4} - \frac{\sqrt{2}}{2} \cdot 1 + 1 (- \frac{\sqrt{2}}{2}) + 1 \cdot 1) (2 (- \frac{\sqrt{2}}{2}) - {(- \frac{\sqrt{2}}{2})}^{2})$

Step 4.3.2.3.1.2.4

Cancel the common factor of $2$ .

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Step 4.3.2.3.1.2.4.1

Cancel the common factor.

$f (- \frac{\sqrt{2}}{2}) = (\frac{2^{\frac{2}{2}}}{4} - \frac{\sqrt{2}}{2} \cdot 1 + 1 (- \frac{\sqrt{2}}{2}) + 1 \cdot 1) (2 (- \frac{\sqrt{2}}{2}) - {(- \frac{\sqrt{2}}{2})}^{2})$

Step 4.3.2.3.1.2.4.2

Rewrite the expression.

$f (- \frac{\sqrt{2}}{2}) = (\frac{2}{4} - \frac{\sqrt{2}}{2} \cdot 1 + 1 (- \frac{\sqrt{2}}{2}) + 1 \cdot 1) (2 (- \frac{\sqrt{2}}{2}) - {(- \frac{\sqrt{2}}{2})}^{2})$

Step 4.3.2.3.1.2.5

Evaluate the exponent.

$f (- \frac{\sqrt{2}}{2}) = (\frac{2}{4} - \frac{\sqrt{2}}{2} \cdot 1 + 1 (- \frac{\sqrt{2}}{2}) + 1 \cdot 1) (2 (- \frac{\sqrt{2}}{2}) - {(- \frac{\sqrt{2}}{2})}^{2})$

Step 4.3.2.3.1.3

Cancel the common factor of $2$ and $4$ .

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Step 4.3.2.3.1.3.1

Factor $2$ out of $2$ .

$f (- \frac{\sqrt{2}}{2}) = (\frac{2 (1)}{4} - \frac{\sqrt{2}}{2} \cdot 1 + 1 (- \frac{\sqrt{2}}{2}) + 1 \cdot 1) (2 (- \frac{\sqrt{2}}{2}) - {(- \frac{\sqrt{2}}{2})}^{2})$

Step 4.3.2.3.1.3.2

Cancel the common factors.

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Step 4.3.2.3.1.3.2.1

Factor $2$ out of $4$ .

$f (- \frac{\sqrt{2}}{2}) = (\frac{2 \cdot 1}{2 \cdot 2} - \frac{\sqrt{2}}{2} \cdot 1 + 1 (- \frac{\sqrt{2}}{2}) + 1 \cdot 1) (2 (- \frac{\sqrt{2}}{2}) - {(- \frac{\sqrt{2}}{2})}^{2})$

Step 4.3.2.3.1.3.2.2

Cancel the common factor.

$f (- \frac{\sqrt{2}}{2}) = (\frac{2 \cdot 1}{2 \cdot 2} - \frac{\sqrt{2}}{2} \cdot 1 + 1 (- \frac{\sqrt{2}}{2}) + 1 \cdot 1) (2 (- \frac{\sqrt{2}}{2}) - {(- \frac{\sqrt{2}}{2})}^{2})$

Step 4.3.2.3.1.3.2.3

Rewrite the expression.

$f (- \frac{\sqrt{2}}{2}) = (\frac{1}{2} - \frac{\sqrt{2}}{2} \cdot 1 + 1 (- \frac{\sqrt{2}}{2}) + 1 \cdot 1) (2 (- \frac{\sqrt{2}}{2}) - {(- \frac{\sqrt{2}}{2})}^{2})$

Step 4.3.2.3.1.4

Multiply $- 1$ by $1$ .

$f (- \frac{\sqrt{2}}{2}) = (\frac{1}{2} - \frac{\sqrt{2}}{2} + 1 (- \frac{\sqrt{2}}{2}) + 1 \cdot 1) (2 (- \frac{\sqrt{2}}{2}) - {(- \frac{\sqrt{2}}{2})}^{2})$

Step 4.3.2.3.1.5

Multiply $- \frac{\sqrt{2}}{2}$ by $1$ .

$f (- \frac{\sqrt{2}}{2}) = (\frac{1}{2} - \frac{\sqrt{2}}{2} - \frac{\sqrt{2}}{2} + 1 \cdot 1) (2 (- \frac{\sqrt{2}}{2}) - {(- \frac{\sqrt{2}}{2})}^{2})$

Step 4.3.2.3.1.6

Multiply $1$ by $1$ .

$f (- \frac{\sqrt{2}}{2}) = (\frac{1}{2} - \frac{\sqrt{2}}{2} - \frac{\sqrt{2}}{2} + 1) (2 (- \frac{\sqrt{2}}{2}) - {(- \frac{\sqrt{2}}{2})}^{2})$

Step 4.3.2.3.2

Write $1$ as a fraction with a common denominator.

$f (- \frac{\sqrt{2}}{2}) = (\frac{1}{2} + \frac{2}{2} - \frac{\sqrt{2}}{2} - \frac{\sqrt{2}}{2}) (2 (- \frac{\sqrt{2}}{2}) - {(- \frac{\sqrt{2}}{2})}^{2})$

Step 4.3.2.3.3

Combine the numerators over the common denominator.

$f (- \frac{\sqrt{2}}{2}) = (\frac{1 + 2}{2} - \frac{\sqrt{2}}{2} - \frac{\sqrt{2}}{2}) (2 (- \frac{\sqrt{2}}{2}) - {(- \frac{\sqrt{2}}{2})}^{2})$

Step 4.3.2.3.4

Add $1$ and $2$ .

$f (- \frac{\sqrt{2}}{2}) = (\frac{3}{2} - \frac{\sqrt{2}}{2} - \frac{\sqrt{2}}{2}) (2 (- \frac{\sqrt{2}}{2}) - {(- \frac{\sqrt{2}}{2})}^{2})$

Step 4.3.2.3.5

Subtract $\frac{\sqrt{2}}{2}$ from $- \frac{\sqrt{2}}{2}$ .

$f (- \frac{\sqrt{2}}{2}) = (\frac{3}{2} - 2 \frac{\sqrt{2}}{2}) (2 (- \frac{\sqrt{2}}{2}) - {(- \frac{\sqrt{2}}{2})}^{2})$

Step 4.3.2.4

Simplify terms.

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Step 4.3.2.4.1

Simplify each term.

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Step 4.3.2.4.1.1

Cancel the common factor of $2$ .

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Step 4.3.2.4.1.1.1

Factor $2$ out of $- 2$ .

$f (- \frac{\sqrt{2}}{2}) = (\frac{3}{2} + 2 (- 1) (\frac{\sqrt{2}}{2})) (2 (- \frac{\sqrt{2}}{2}) - {(- \frac{\sqrt{2}}{2})}^{2})$

Step 4.3.2.4.1.1.2

Cancel the common factor.

$f (- \frac{\sqrt{2}}{2}) = (\frac{3}{2} + 2 \cdot (- 1 \frac{\sqrt{2}}{2})) (2 (- \frac{\sqrt{2}}{2}) - {(- \frac{\sqrt{2}}{2})}^{2})$

Step 4.3.2.4.1.1.3

Rewrite the expression.

$f (- \frac{\sqrt{2}}{2}) = (\frac{3}{2} - 1 \sqrt{2}) (2 (- \frac{\sqrt{2}}{2}) - {(- \frac{\sqrt{2}}{2})}^{2})$

Step 4.3.2.4.1.2

Rewrite $- 1 \sqrt{2}$ as $- \sqrt{2}$ .

$f (- \frac{\sqrt{2}}{2}) = (\frac{3}{2} - \sqrt{2}) (2 (- \frac{\sqrt{2}}{2}) - {(- \frac{\sqrt{2}}{2})}^{2})$

Step 4.3.2.4.2

Simplify each term.

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Step 4.3.2.4.2.1

Cancel the common factor of $2$ .

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Step 4.3.2.4.2.1.1

Move the leading negative in $- \frac{\sqrt{2}}{2}$ into the numerator.

$f (- \frac{\sqrt{2}}{2}) = (\frac{3}{2} - \sqrt{2}) (2 (\frac{- \sqrt{2}}{2}) - {(- \frac{\sqrt{2}}{2})}^{2})$

Step 4.3.2.4.2.1.2

Cancel the common factor.

$f (- \frac{\sqrt{2}}{2}) = (\frac{3}{2} - \sqrt{2}) (2 (\frac{- \sqrt{2}}{2}) - {(- \frac{\sqrt{2}}{2})}^{2})$

Step 4.3.2.4.2.1.3

Rewrite the expression.

$f (- \frac{\sqrt{2}}{2}) = (\frac{3}{2} - \sqrt{2}) (- \sqrt{2} - {(- \frac{\sqrt{2}}{2})}^{2})$

Step 4.3.2.4.2.2

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 4.3.2.4.2.2.1

Apply the product rule to $- \frac{\sqrt{2}}{2}$ .

$f (- \frac{\sqrt{2}}{2}) = (\frac{3}{2} - \sqrt{2}) (- \sqrt{2} - ({(- 1)}^{2} {(\frac{\sqrt{2}}{2})}^{2}))$

Step 4.3.2.4.2.2.2

Apply the product rule to $\frac{\sqrt{2}}{2}$ .

$f (- \frac{\sqrt{2}}{2}) = (\frac{3}{2} - \sqrt{2}) (- \sqrt{2} - ({(- 1)}^{2} (\frac{{\sqrt{2}}^{2}}{2^{2}})))$

Step 4.3.2.4.2.3

Multiply $- 1$ by ${(- 1)}^{2}$ by adding the exponents.

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Step 4.3.2.4.2.3.1

Move ${(- 1)}^{2}$ .

$f (- \frac{\sqrt{2}}{2}) = (\frac{3}{2} - \sqrt{2}) (- \sqrt{2} + {(- 1)}^{2} \cdot (- 1 \frac{{\sqrt{2}}^{2}}{2^{2}}))$

Step 4.3.2.4.2.3.2

Multiply ${(- 1)}^{2}$ by $- 1$ .

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Step 4.3.2.4.2.3.2.1

Raise $- 1$ to the power of $1$ .

$f (- \frac{\sqrt{2}}{2}) = (\frac{3}{2} - \sqrt{2}) (- \sqrt{2} + {(- 1)}^{2} \cdot ((- 1) (\frac{{\sqrt{2}}^{2}}{2^{2}})))$

Step 4.3.2.4.2.3.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f (- \frac{\sqrt{2}}{2}) = (\frac{3}{2} - \sqrt{2}) (- \sqrt{2} + {(- 1)}^{2 + 1} (\frac{{\sqrt{2}}^{2}}{2^{2}}))$

Step 4.3.2.4.2.3.3

Add $2$ and $1$ .

$f (- \frac{\sqrt{2}}{2}) = (\frac{3}{2} - \sqrt{2}) (- \sqrt{2} + {(- 1)}^{3} (\frac{{\sqrt{2}}^{2}}{2^{2}}))$

Step 4.3.2.4.2.4

Raise $- 1$ to the power of $3$ .

$f (- \frac{\sqrt{2}}{2}) = (\frac{3}{2} - \sqrt{2}) (- \sqrt{2} - \frac{{\sqrt{2}}^{2}}{2^{2}})$

Step 4.3.2.4.2.5

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

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Step 4.3.2.4.2.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

$f (- \frac{\sqrt{2}}{2}) = (\frac{3}{2} - \sqrt{2}) (- \sqrt{2} - \frac{{(2^{\frac{1}{2}})}^{2}}{2^{2}})$

Step 4.3.2.4.2.5.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$f (- \frac{\sqrt{2}}{2}) = (\frac{3}{2} - \sqrt{2}) (- \sqrt{2} - \frac{2^{\frac{1}{2} \cdot 2}}{2^{2}})$

Step 4.3.2.4.2.5.3

Combine $\frac{1}{2}$ and $2$ .

$f (- \frac{\sqrt{2}}{2}) = (\frac{3}{2} - \sqrt{2}) (- \sqrt{2} - \frac{2^{\frac{2}{2}}}{2^{2}})$

Step 4.3.2.4.2.5.4

Cancel the common factor of $2$ .

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Step 4.3.2.4.2.5.4.1

Cancel the common factor.

$f (- \frac{\sqrt{2}}{2}) = (\frac{3}{2} - \sqrt{2}) (- \sqrt{2} - \frac{2^{\frac{2}{2}}}{2^{2}})$

Step 4.3.2.4.2.5.4.2

Rewrite the expression.

$f (- \frac{\sqrt{2}}{2}) = (\frac{3}{2} - \sqrt{2}) (- \sqrt{2} - \frac{2}{2^{2}})$

Step 4.3.2.4.2.5.5

Evaluate the exponent.

$f (- \frac{\sqrt{2}}{2}) = (\frac{3}{2} - \sqrt{2}) (- \sqrt{2} - \frac{2}{2^{2}})$

Step 4.3.2.4.2.6

Raise $2$ to the power of $2$ .

$f (- \frac{\sqrt{2}}{2}) = (\frac{3}{2} - \sqrt{2}) (- \sqrt{2} - \frac{2}{4})$

Step 4.3.2.4.2.7

Cancel the common factor of $2$ and $4$ .

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Step 4.3.2.4.2.7.1

Factor $2$ out of $2$ .

$f (- \frac{\sqrt{2}}{2}) = (\frac{3}{2} - \sqrt{2}) (- \sqrt{2} - \frac{2 (1)}{4})$

Step 4.3.2.4.2.7.2

Cancel the common factors.

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Step 4.3.2.4.2.7.2.1

Factor $2$ out of $4$ .

$f (- \frac{\sqrt{2}}{2}) = (\frac{3}{2} - \sqrt{2}) (- \sqrt{2} - \frac{2 \cdot 1}{2 \cdot 2})$

Step 4.3.2.4.2.7.2.2

Cancel the common factor.

$f (- \frac{\sqrt{2}}{2}) = (\frac{3}{2} - \sqrt{2}) (- \sqrt{2} - \frac{2 \cdot 1}{2 \cdot 2})$

Step 4.3.2.4.2.7.2.3

Rewrite the expression.

$f (- \frac{\sqrt{2}}{2}) = (\frac{3}{2} - \sqrt{2}) (- \sqrt{2} - \frac{1}{2})$

Step 4.3.2.5

Expand $(\frac{3}{2} - \sqrt{2}) (- \sqrt{2} - \frac{1}{2})$ using the FOIL Method.

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Step 4.3.2.5.1

Apply the distributive property.

$f (- \frac{\sqrt{2}}{2}) = \frac{3}{2} \cdot (- \sqrt{2} - \frac{1}{2}) - \sqrt{2} (- \sqrt{2} - \frac{1}{2})$

Step 4.3.2.5.2

Apply the distributive property.

$f (- \frac{\sqrt{2}}{2}) = \frac{3}{2} \cdot (- \sqrt{2}) + \frac{3}{2} \cdot (- \frac{1}{2}) - \sqrt{2} (- \sqrt{2} - \frac{1}{2})$

Step 4.3.2.5.3

Apply the distributive property.

$f (- \frac{\sqrt{2}}{2}) = \frac{3}{2} \cdot (- \sqrt{2}) + \frac{3}{2} \cdot (- \frac{1}{2}) - \sqrt{2} (- \sqrt{2}) - \sqrt{2} (- \frac{1}{2})$

Step 4.3.2.6

Simplify and combine like terms.

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Step 4.3.2.6.1

Simplify each term.

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Step 4.3.2.6.1.1

Combine $\frac{3}{2}$ and $\sqrt{2}$ .

$f (- \frac{\sqrt{2}}{2}) = - \frac{3 \sqrt{2}}{2} + \frac{3}{2} \cdot (- \frac{1}{2}) - \sqrt{2} (- \sqrt{2}) - \sqrt{2} (- \frac{1}{2})$

Step 4.3.2.6.1.2

Multiply $\frac{3}{2} (- \frac{1}{2})$ .

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Step 4.3.2.6.1.2.1

Multiply $\frac{3}{2}$ by $\frac{1}{2}$ .

$f (- \frac{\sqrt{2}}{2}) = - \frac{3 \sqrt{2}}{2} - \frac{3}{2 \cdot 2} - \sqrt{2} (- \sqrt{2}) - \sqrt{2} (- \frac{1}{2})$

Step 4.3.2.6.1.2.2

Multiply $2$ by $2$ .

$f (- \frac{\sqrt{2}}{2}) = - \frac{3 \sqrt{2}}{2} - \frac{3}{4} - \sqrt{2} (- \sqrt{2}) - \sqrt{2} (- \frac{1}{2})$

Step 4.3.2.6.1.3

Multiply $- \sqrt{2} (- \sqrt{2})$ .

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Step 4.3.2.6.1.3.1

Multiply $- 1$ by $- 1$ .

$f (- \frac{\sqrt{2}}{2}) = - \frac{3 \sqrt{2}}{2} - \frac{3}{4} + 1 \sqrt{2} \sqrt{2} - \sqrt{2} (- \frac{1}{2})$

Step 4.3.2.6.1.3.2

Multiply $\sqrt{2}$ by $1$ .

$f (- \frac{\sqrt{2}}{2}) = - \frac{3 \sqrt{2}}{2} - \frac{3}{4} + \sqrt{2} \sqrt{2} - \sqrt{2} (- \frac{1}{2})$

Step 4.3.2.6.1.3.3

Raise $\sqrt{2}$ to the power of $1$ .

$f (- \frac{\sqrt{2}}{2}) = - \frac{3 \sqrt{2}}{2} - \frac{3}{4} + \sqrt{2} \sqrt{2} - \sqrt{2} (- \frac{1}{2})$

Step 4.3.2.6.1.3.4

Raise $\sqrt{2}$ to the power of $1$ .

$f (- \frac{\sqrt{2}}{2}) = - \frac{3 \sqrt{2}}{2} - \frac{3}{4} + \sqrt{2} \sqrt{2} - \sqrt{2} (- \frac{1}{2})$

Step 4.3.2.6.1.3.5

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f (- \frac{\sqrt{2}}{2}) = - \frac{3 \sqrt{2}}{2} - \frac{3}{4} + {\sqrt{2}}^{1 + 1} - \sqrt{2} (- \frac{1}{2})$

Step 4.3.2.6.1.3.6

Add $1$ and $1$ .

$f (- \frac{\sqrt{2}}{2}) = - \frac{3 \sqrt{2}}{2} - \frac{3}{4} + {\sqrt{2}}^{2} - \sqrt{2} (- \frac{1}{2})$

Step 4.3.2.6.1.4

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

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Step 4.3.2.6.1.4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

$f (- \frac{\sqrt{2}}{2}) = - \frac{3 \sqrt{2}}{2} - \frac{3}{4} + {(2^{\frac{1}{2}})}^{2} - \sqrt{2} (- \frac{1}{2})$

Step 4.3.2.6.1.4.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$f (- \frac{\sqrt{2}}{2}) = - \frac{3 \sqrt{2}}{2} - \frac{3}{4} + 2^{\frac{1}{2} \cdot 2} - \sqrt{2} (- \frac{1}{2})$

Step 4.3.2.6.1.4.3

Combine $\frac{1}{2}$ and $2$ .

$f (- \frac{\sqrt{2}}{2}) = - \frac{3 \sqrt{2}}{2} - \frac{3}{4} + 2^{\frac{2}{2}} - \sqrt{2} (- \frac{1}{2})$

Step 4.3.2.6.1.4.4

Cancel the common factor of $2$ .

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Step 4.3.2.6.1.4.4.1

Cancel the common factor.

$f (- \frac{\sqrt{2}}{2}) = - \frac{3 \sqrt{2}}{2} - \frac{3}{4} + 2^{\frac{2}{2}} - \sqrt{2} (- \frac{1}{2})$

Step 4.3.2.6.1.4.4.2

Rewrite the expression.

$f (- \frac{\sqrt{2}}{2}) = - \frac{3 \sqrt{2}}{2} - \frac{3}{4} + 2 - \sqrt{2} (- \frac{1}{2})$

Step 4.3.2.6.1.4.5

Evaluate the exponent.

$f (- \frac{\sqrt{2}}{2}) = - \frac{3 \sqrt{2}}{2} - \frac{3}{4} + 2 - \sqrt{2} (- \frac{1}{2})$

Step 4.3.2.6.1.5

Multiply $- \sqrt{2} (- \frac{1}{2})$ .

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Step 4.3.2.6.1.5.1

Multiply $- 1$ by $- 1$ .

$f (- \frac{\sqrt{2}}{2}) = - \frac{3 \sqrt{2}}{2} - \frac{3}{4} + 2 + 1 \sqrt{2} (\frac{1}{2})$

Step 4.3.2.6.1.5.2

Multiply $\sqrt{2}$ by $1$ .

$f (- \frac{\sqrt{2}}{2}) = - \frac{3 \sqrt{2}}{2} - \frac{3}{4} + 2 + \sqrt{2} (\frac{1}{2})$

Step 4.3.2.6.1.5.3

Combine $\sqrt{2}$ and $\frac{1}{2}$ .

$f (- \frac{\sqrt{2}}{2}) = - \frac{3 \sqrt{2}}{2} - \frac{3}{4} + 2 + \frac{\sqrt{2}}{2}$

Step 4.3.2.6.2

Combine the numerators over the common denominator.

$f (- \frac{\sqrt{2}}{2}) = \frac{- 3 \sqrt{2} + \sqrt{2}}{2} - \frac{3}{4} + 2$

Step 4.3.2.6.3

To write $2$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$f (- \frac{\sqrt{2}}{2}) = \frac{- 3 \sqrt{2} + \sqrt{2}}{2} - \frac{3}{4} + 2 \cdot \frac{4}{4}$

Step 4.3.2.6.4

Combine $2$ and $\frac{4}{4}$ .

$f (- \frac{\sqrt{2}}{2}) = \frac{- 3 \sqrt{2} + \sqrt{2}}{2} - \frac{3}{4} + \frac{2 \cdot 4}{4}$

Step 4.3.2.6.5

Combine the numerators over the common denominator.

$f (- \frac{\sqrt{2}}{2}) = \frac{- 3 \sqrt{2} + \sqrt{2}}{2} + \frac{- 3 + 2 \cdot 4}{4}$

Step 4.3.2.6.6

Simplify the numerator.

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Step 4.3.2.6.6.1

Multiply $2$ by $4$ .

$f (- \frac{\sqrt{2}}{2}) = \frac{- 3 \sqrt{2} + \sqrt{2}}{2} + \frac{- 3 + 8}{4}$

Step 4.3.2.6.6.2

Add $- 3$ and $8$ .

$f (- \frac{\sqrt{2}}{2}) = \frac{- 3 \sqrt{2} + \sqrt{2}}{2} + \frac{5}{4}$

Step 4.3.2.7

Add $- 3 \sqrt{2}$ and $\sqrt{2}$ .

$f (- \frac{\sqrt{2}}{2}) = \frac{- 2 \sqrt{2}}{2} + \frac{5}{4}$

Step 4.3.2.8

Cancel the common factor of $- 2$ and $2$ .

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Step 4.3.2.8.1

Factor $2$ out of $- 2 \sqrt{2}$ .

$f (- \frac{\sqrt{2}}{2}) = \frac{2 (- \sqrt{2})}{2} + \frac{5}{4}$

Step 4.3.2.8.2

Cancel the common factors.

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Step 4.3.2.8.2.1

Factor $2$ out of $2$ .

$f (- \frac{\sqrt{2}}{2}) = \frac{2 (- \sqrt{2})}{2 (1)} + \frac{5}{4}$

Step 4.3.2.8.2.2

Cancel the common factor.

$f (- \frac{\sqrt{2}}{2}) = \frac{2 (- \sqrt{2})}{2 \cdot 1} + \frac{5}{4}$

Step 4.3.2.8.2.3

Rewrite the expression.

$f (- \frac{\sqrt{2}}{2}) = \frac{- \sqrt{2}}{1} + \frac{5}{4}$

Step 4.3.2.8.2.4

Divide $- \sqrt{2}$ by $1$ .

$f (- \frac{\sqrt{2}}{2}) = - \sqrt{2} + \frac{5}{4}$

Step 4.3.2.9

The final answer is $- \sqrt{2} + \frac{5}{4}$ .

$- \sqrt{2} + \frac{5}{4}$

Step 4.4

The point found by substituting $- \frac{\sqrt{2}}{2}$ in $f (x) = {(x + 1)}^{2} (2 x - x^{2})$ is $(- \frac{\sqrt{2}}{2}, - \sqrt{2} + \frac{5}{4})$ . This point can be an inflection point.

$(- \frac{\sqrt{2}}{2}, - \sqrt{2} + \frac{5}{4})$

Step 4.5

Determine the points that could be inflection points.

$(\frac{\sqrt{2}}{2}, \frac{5 + 4 \sqrt{2}}{4}), (- \frac{\sqrt{2}}{2}, - \sqrt{2} + \frac{5}{4})$

Step 5

Split $(- \infty, \infty)$ into intervals around the points that could potentially be inflection points.

$(- \infty, - \frac{\sqrt{2}}{2}) \cup (- \frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}) \cup (\frac{\sqrt{2}}{2}, \infty)$

Step 6

Substitute a value from the interval $(- \infty, - \frac{\sqrt{2}}{2})$ into the second derivative to determine if it is increasing or decreasing.

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Step 6.1

Replace the variable $x$ with $- 0.80710678$ in the expression.

$f'' (- 0.80710678) = - 12 {(- 0.80710678)}^{2} + 6$

Step 6.2

Simplify the result.

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Step 6.2.1

Simplify each term.

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Step 6.2.1.1

Raise $- 0.80710678$ to the power of $2$ .

$f'' (- 0.80710678) = - 12 \cdot 0.65142135 + 6$

Step 6.2.1.2

Multiply $- 12$ by $0.65142135$ .

$f'' (- 0.80710678) = - 7.81705627 + 6$

Step 6.2.2

Add $- 7.81705627$ and $6$ .

$f'' (- 0.80710678) = - 1.81705627$

Step 6.2.3

The final answer is $- 1.81705627$ .

$- 1.81705627$

Step 6.3

At $- 0.80710678$ , the second derivative is $- 1.81705627$ . Since this is negative, the second derivative is decreasing on the interval $(- \infty, - \frac{\sqrt{2}}{2})$

Decreasing on $(- \infty, - \frac{\sqrt{2}}{2})$ since $f'' (x) < 0$

Step 7

Substitute a value from the interval $(- \frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2})$ into the second derivative to determine if it is increasing or decreasing.

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Step 7.1

Replace the variable $x$ with $0$ in the expression.

$f'' (0) = - 12 {(0)}^{2} + 6$

Step 7.2

Simplify the result.

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Step 7.2.1

Simplify each term.

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Step 7.2.1.1

Raising $0$ to any positive power yields $0$ .

$f'' (0) = - 12 \cdot 0 + 6$

Step 7.2.1.2

Multiply $- 12$ by $0$ .

$f'' (0) = 0 + 6$

Step 7.2.2

Add $0$ and $6$ .

$f'' (0) = 6$

Step 7.2.3

The final answer is $6$ .

$6$

Step 7.3

At $0$ , the second derivative is $6$ . Since this is positive, the second derivative is increasing on the interval $(- \frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2})$ .

Increasing on $(- \frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2})$ since $f'' (x) > 0$

Step 8

Substitute a value from the interval $(\frac{\sqrt{2}}{2}, \infty)$ into the second derivative to determine if it is increasing or decreasing.

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Step 8.1

Replace the variable $x$ with $0.80710678$ in the expression.

$f'' (0.80710678) = - 12 {(0.80710678)}^{2} + 6$

Step 8.2

Simplify the result.

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Step 8.2.1

Simplify each term.

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Step 8.2.1.1

Raise $0.80710678$ to the power of $2$ .

$f'' (0.80710678) = - 12 \cdot 0.65142135 + 6$

Step 8.2.1.2

Multiply $- 12$ by $0.65142135$ .

$f'' (0.80710678) = - 7.81705627 + 6$

Step 8.2.2

Add $- 7.81705627$ and $6$ .

$f'' (0.80710678) = - 1.81705627$

Step 8.2.3

The final answer is $- 1.81705627$ .

$- 1.81705627$

Step 8.3

At $0.80710678$ , the second derivative is $- 1.81705627$ . Since this is negative, the second derivative is decreasing on the interval $(\frac{\sqrt{2}}{2}, \infty)$

Decreasing on $(\frac{\sqrt{2}}{2}, \infty)$ since $f'' (x) < 0$

Step 9

An inflection point is a point on a curve at which the concavity changes sign from plus to minus or from minus to plus. The inflection points in this case are $(- \frac{\sqrt{2}}{2}, - \sqrt{2} + \frac{5}{4}), (\frac{\sqrt{2}}{2}, \frac{5 + 4 \sqrt{2}}{4})$ .

$(- \frac{\sqrt{2}}{2}, - \sqrt{2} + \frac{5}{4}), (\frac{\sqrt{2}}{2}, \frac{5 + 4 \sqrt{2}}{4})$

Step 10