Calculus Examples

$f (x) = {(1 - 2 x)}^{3}$

Step 1

Find the $x$ values where the second derivative is equal to $0$ .

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Step 1.1

Find the second derivative.

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Step 1.1.1

Find the first derivative.

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Step 1.1.1.1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{3}$ and $g (x) = 1 - 2 x$ .

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Step 1.1.1.1.1

To apply the Chain Rule, set $u$ as $1 - 2 x$ .

$\frac{d}{d u} [u^{3}] \frac{d}{d x} [1 - 2 x]$

Step 1.1.1.1.2

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

$3 u^{2} \frac{d}{d x} [1 - 2 x]$

Step 1.1.1.1.3

Replace all occurrences of $u$ with $1 - 2 x$ .

$3 {(1 - 2 x)}^{2} \frac{d}{d x} [1 - 2 x]$

Step 1.1.1.2

Differentiate.

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Step 1.1.1.2.1

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

$3 {(1 - 2 x)}^{2} (\frac{d}{d x} [1] + \frac{d}{d x} [- 2 x])$

Step 1.1.1.2.2

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

$3 {(1 - 2 x)}^{2} (0 + \frac{d}{d x} [- 2 x])$

Step 1.1.1.2.3

Add $0$ and $\frac{d}{d x} [- 2 x]$ .

$3 {(1 - 2 x)}^{2} \frac{d}{d x} [- 2 x]$

Step 1.1.1.2.4

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

$3 {(1 - 2 x)}^{2} (- 2 \frac{d}{d x} [x])$

Step 1.1.1.2.5

Multiply $- 2$ by $3$ .

$- 6 {(1 - 2 x)}^{2} \frac{d}{d x} [x]$

Step 1.1.1.2.6

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

$- 6 {(1 - 2 x)}^{2} \cdot 1$

Step 1.1.1.2.7

Multiply $- 6$ by $1$ .

$f' (x) = - 6 {(1 - 2 x)}^{2}$

Step 1.1.2

Find the second derivative.

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Step 1.1.2.1

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

$\frac{d}{d x} [- 6 ((1 - 2 x) (1 - 2 x))]$

Step 1.1.2.2

Expand $(1 - 2 x) (1 - 2 x)$ using the FOIL Method.

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Step 1.1.2.2.1

Apply the distributive property.

$\frac{d}{d x} [- 6 (1 (1 - 2 x) - 2 x (1 - 2 x))]$

Step 1.1.2.2.2

Apply the distributive property.

$\frac{d}{d x} [- 6 (1 \cdot 1 + 1 (- 2 x) - 2 x (1 - 2 x))]$

Step 1.1.2.2.3

Apply the distributive property.

$\frac{d}{d x} [- 6 (1 \cdot 1 + 1 (- 2 x) - 2 x \cdot 1 - 2 x (- 2 x))]$

Step 1.1.2.3

Simplify and combine like terms.

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Step 1.1.2.3.1

Simplify each term.

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Step 1.1.2.3.1.1

Multiply $1$ by $1$ .

$\frac{d}{d x} [- 6 (1 + 1 (- 2 x) - 2 x \cdot 1 - 2 x (- 2 x))]$

Step 1.1.2.3.1.2

Multiply $- 2 x$ by $1$ .

$\frac{d}{d x} [- 6 (1 - 2 x - 2 x \cdot 1 - 2 x (- 2 x))]$

Step 1.1.2.3.1.3

Multiply $- 2$ by $1$ .

$\frac{d}{d x} [- 6 (1 - 2 x - 2 x - 2 x (- 2 x))]$

Step 1.1.2.3.1.4

Rewrite using the commutative property of multiplication.

$\frac{d}{d x} [- 6 (1 - 2 x - 2 x - 2 \cdot - 2 x \cdot x)]$

Step 1.1.2.3.1.5

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

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Step 1.1.2.3.1.5.1

Move $x$ .

$\frac{d}{d x} [- 6 (1 - 2 x - 2 x - 2 \cdot - 2 (x \cdot x))]$

Step 1.1.2.3.1.5.2

Multiply $x$ by $x$ .

$\frac{d}{d x} [- 6 (1 - 2 x - 2 x - 2 \cdot - 2 x^{2})]$

Step 1.1.2.3.1.6

Multiply $- 2$ by $- 2$ .

$\frac{d}{d x} [- 6 (1 - 2 x - 2 x + 4 x^{2})]$

Step 1.1.2.3.2

Subtract $2 x$ from $- 2 x$ .

$\frac{d}{d x} [- 6 (1 - 4 x + 4 x^{2})]$

Step 1.1.2.4

Since $- 6$ is constant with respect to $x$ , the derivative of $- 6 (1 - 4 x + 4 x^{2})$ with respect to $x$ is $- 6 \frac{d}{d x} [1 - 4 x + 4 x^{2}]$ .

$- 6 \frac{d}{d x} [1 - 4 x + 4 x^{2}]$

Step 1.1.2.5

By the Sum Rule, the derivative of $1 - 4 x + 4 x^{2}$ with respect to $x$ is $\frac{d}{d x} [1] + \frac{d}{d x} [- 4 x] + \frac{d}{d x} [4 x^{2}]$ .

$- 6 (\frac{d}{d x} [1] + \frac{d}{d x} [- 4 x] + \frac{d}{d x} [4 x^{2}])$

Step 1.1.2.6

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

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

Step 1.1.2.7

Add $0$ and $\frac{d}{d x} [- 4 x]$ .

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

Step 1.1.2.8

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

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

Step 1.1.2.9

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

$- 6 (- 4 \cdot 1 + \frac{d}{d x} [4 x^{2}])$

Step 1.1.2.10

Multiply $- 4$ by $1$ .

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

Step 1.1.2.11

Since $4$ is constant with respect to $x$ , the derivative of $4 x^{2}$ with respect to $x$ is $4 \frac{d}{d x} [x^{2}]$ .

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

Step 1.1.2.12

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

$- 6 (- 4 + 4 (2 x))$

Step 1.1.2.13

Multiply $2$ by $4$ .

$- 6 (- 4 + 8 x)$

Step 1.1.2.14

Simplify.

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Step 1.1.2.14.1

Apply the distributive property.

$- 6 \cdot - 4 - 6 (8 x)$

Step 1.1.2.14.2

Combine terms.

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Step 1.1.2.14.2.1

Multiply $- 6$ by $- 4$ .

$24 - 6 (8 x)$

Step 1.1.2.14.2.2

Multiply $8$ by $- 6$ .

$24 - 48 x$

Step 1.1.2.14.3

Reorder terms.

$f'' (x) = - 48 x + 24$

Step 1.1.3

The second derivative of $f (x)$ with respect to $x$ is $- 48 x + 24$ .

$- 48 x + 24$

Step 1.2

Set the second derivative equal to $0$ then solve the equation $- 48 x + 24 = 0$ .

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Step 1.2.1

Set the second derivative equal to $0$ .

$- 48 x + 24 = 0$

Step 1.2.2

Subtract $24$ from both sides of the equation.

$- 48 x = - 24$

Step 1.2.3

Divide each term in $- 48 x = - 24$ by $- 48$ and simplify.

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Step 1.2.3.1

Divide each term in $- 48 x = - 24$ by $- 48$ .

$\frac{- 48 x}{- 48} = \frac{- 24}{- 48}$

Step 1.2.3.2

Simplify the left side.

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Step 1.2.3.2.1

Cancel the common factor of $- 48$ .

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Step 1.2.3.2.1.1

Cancel the common factor.

$\frac{- 48 x}{- 48} = \frac{- 24}{- 48}$

Step 1.2.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 24}{- 48}$

Step 1.2.3.3

Simplify the right side.

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Step 1.2.3.3.1

Cancel the common factor of $- 24$ and $- 48$ .

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Step 1.2.3.3.1.1

Factor $- 24$ out of $- 24$ .

$x = \frac{- 24 (1)}{- 48}$

Step 1.2.3.3.1.2

Cancel the common factors.

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Step 1.2.3.3.1.2.1

Factor $- 24$ out of $- 48$ .

$x = \frac{- 24 \cdot 1}{- 24 \cdot 2}$

Step 1.2.3.3.1.2.2

Cancel the common factor.

$x = \frac{- 24 \cdot 1}{- 24 \cdot 2}$

Step 1.2.3.3.1.2.3

Rewrite the expression.

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

Step 2

The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.

Interval Notation:

$(- \infty, \infty)$

Set-Builder Notation:

${x | x \in ℝ}$

Step 3

Create intervals around the $x$ -values where the second derivative is zero or undefined.

$(- \infty, \frac{1}{2}) \cup (\frac{1}{2}, \infty)$

Step 4

Substitute any number from the interval $(- \infty, \frac{1}{2})$ into the second derivative and evaluate to determine the concavity.

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Step 4.1

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

$f'' (0) = - 48 \cdot 0 + 24$

Step 4.2

Simplify the result.

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Step 4.2.1

Multiply $- 48$ by $0$ .

$f'' (0) = 0 + 24$

Step 4.2.2

Add $0$ and $24$ .

$f'' (0) = 24$

Step 4.2.3

The final answer is $24$ .

$24$

Step 4.3

The graph is concave up on the interval $(- \infty, \frac{1}{2})$ because $f'' (0)$ is positive.

The graph is concave up

Step 5

Substitute any number from the interval $(\frac{1}{2}, \infty)$ into the second derivative and evaluate to determine the concavity.

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Step 5.1

Replace the variable $x$ with $3$ in the expression.

$f'' (3) = - 48 \cdot 3 + 24$

Step 5.2

Simplify the result.

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Step 5.2.1

Multiply $- 48$ by $3$ .

$f'' (3) = - 144 + 24$

Step 5.2.2

Add $- 144$ and $24$ .

$f'' (3) = - 120$

Step 5.2.3

The final answer is $- 120$ .

$- 120$

Step 5.3

The graph is concave down on the interval $(\frac{1}{2}, \infty)$ because $f'' (3)$ is negative.

The graph is concave down

Step 6

The graph is concave down when the second derivative is negative and concave up when the second derivative is positive.

The graph is concave up

The graph is concave down

Step 7