Calculus Examples

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

Step 1

Find the second derivative.

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Step 1.1

Find the first derivative.

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Step 1.1.1

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

$f' (x) = 3 \frac{d}{d x} (x {(x - 3)}^{3})$

Step 1.1.2

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 $f (x) = x$ and $g (x) = {(x - 3)}^{3}$ .

$f' (x) = 3 (x \frac{d}{d x} ({(x - 3)}^{3}) + {(x - 3)}^{3} \frac{d}{d x} (x))$

Step 1.1.3

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 $g (x) = x - 3$ .

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Step 1.1.3.1

To apply the Chain Rule, set $u$ as $x - 3$ .

$f' (x) = 3 (x (\frac{d}{d u} (u^{3}) \frac{d}{d x} (x - 3)) + {(x - 3)}^{3} \frac{d}{d x} (x))$

Step 1.1.3.2

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

$f' (x) = 3 (x (3 u^{2} \frac{d}{d x} (x - 3)) + {(x - 3)}^{3} \frac{d}{d x} (x))$

Step 1.1.3.3

Replace all occurrences of $u$ with $x - 3$ .

$f' (x) = 3 (x (3 {(x - 3)}^{2} \frac{d}{d x} (x - 3)) + {(x - 3)}^{3} \frac{d}{d x} (x))$

Step 1.1.4

Differentiate.

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Step 1.1.4.1

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

$f' (x) = 3 (x (3 {(x - 3)}^{2} (\frac{d}{d x} (x) + \frac{d}{d x} (- 3))) + {(x - 3)}^{3} \frac{d}{d x} (x))$

Step 1.1.4.2

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

$f' (x) = 3 (x (3 {(x - 3)}^{2} (1 + \frac{d}{d x} (- 3))) + {(x - 3)}^{3} \frac{d}{d x} (x))$

Step 1.1.4.3

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3$ with respect to $x$ is $0$ .

$f' (x) = 3 (x (3 {(x - 3)}^{2} (1 + 0)) + {(x - 3)}^{3} \frac{d}{d x} (x))$

Step 1.1.4.4

Simplify the expression.

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Step 1.1.4.4.1

Add $1$ and $0$ .

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

Step 1.1.4.4.2

Multiply $3$ by $1$ .

$f' (x) = 3 (x (3 {(x - 3)}^{2}) + {(x - 3)}^{3} \frac{d}{d x} (x))$

Step 1.1.4.5

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

$f' (x) = 3 (x (3 {(x - 3)}^{2}) + {(x - 3)}^{3} \cdot 1)$

Step 1.1.4.6

Multiply ${(x - 3)}^{3}$ by $1$ .

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

Step 1.1.5

Simplify.

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Step 1.1.5.1

Apply the distributive property.

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

Step 1.1.5.2

Multiply $3$ by $3$ .

$f' (x) = 9 (x ({(x - 3)}^{2})) + 3 {(x - 3)}^{3}$

Step 1.1.5.3

Factor $3 {(x - 3)}^{2}$ out of $9 x {(x - 3)}^{2} + 3 {(x - 3)}^{3}$ .

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Step 1.1.5.3.1

Factor $3 {(x - 3)}^{2}$ out of $9 x {(x - 3)}^{2}$ .

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

Step 1.1.5.3.2

Factor $3 {(x - 3)}^{2}$ out of $3 {(x - 3)}^{3}$ .

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

Step 1.1.5.3.3

Factor $3 {(x - 3)}^{2}$ out of $3 {(x - 3)}^{2} (3 x) + 3 {(x - 3)}^{2} (x - 3)$ .

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

Step 1.1.5.4

Add $3 x$ and $x$ .

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

Step 1.1.5.5

Rewrite ${(x - 3)}^{2}$ as $(x - 3) (x - 3)$ .

$f' (x) = 3 ((x - 3) (x - 3)) (4 x - 3)$

Step 1.1.5.6

Expand $(x - 3) (x - 3)$ using the FOIL Method.

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Step 1.1.5.6.1

Apply the distributive property.

$f' (x) = 3 (x (x - 3) - 3 (x - 3)) (4 x - 3)$

Step 1.1.5.6.2

Apply the distributive property.

$f' (x) = 3 (x \cdot x + x \cdot - 3 - 3 (x - 3)) (4 x - 3)$

Step 1.1.5.6.3

Apply the distributive property.

$f' (x) = 3 (x \cdot x + x \cdot - 3 - 3 x - 3 \cdot - 3) (4 x - 3)$

Step 1.1.5.7

Simplify and combine like terms.

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Step 1.1.5.7.1

Simplify each term.

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Step 1.1.5.7.1.1

Multiply $x$ by $x$ .

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

Step 1.1.5.7.1.2

Move $- 3$ to the left of $x$ .

$f' (x) = 3 (x^{2} - 3 \cdot x - 3 x - 3 \cdot - 3) (4 x - 3)$

Step 1.1.5.7.1.3

Multiply $- 3$ by $- 3$ .

$f' (x) = 3 (x^{2} - 3 x - 3 x + 9) (4 x - 3)$

Step 1.1.5.7.2

Subtract $3 x$ from $- 3 x$ .

$f' (x) = 3 (x^{2} - 6 x + 9) (4 x - 3)$

Step 1.1.5.8

Apply the distributive property.

$f' (x) = (3 x^{2} + 3 (- 6 x) + 3 \cdot 9) (4 x - 3)$

Step 1.1.5.9

Simplify.

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Step 1.1.5.9.1

Multiply $- 6$ by $3$ .

$f' (x) = (3 x^{2} - 18 x + 3 \cdot 9) (4 x - 3)$

Step 1.1.5.9.2

Multiply $3$ by $9$ .

$f' (x) = (3 x^{2} - 18 x + 27) (4 x - 3)$

Step 1.1.5.10

Expand $(3 x^{2} - 18 x + 27) (4 x - 3)$ by multiplying each term in the first expression by each term in the second expression.

$f' (x) = 3 x^{2} (4 x) + 3 x^{2} \cdot - 3 - 18 x (4 x) - 18 x \cdot - 3 + 27 (4 x) + 27 \cdot - 3$

Step 1.1.5.11

Simplify each term.

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Step 1.1.5.11.1

Rewrite using the commutative property of multiplication.

$f' (x) = 3 \cdot (4 x^{2} x) + 3 x^{2} \cdot - 3 - 18 x (4 x) - 18 x \cdot - 3 + 27 (4 x) + 27 \cdot - 3$

Step 1.1.5.11.2

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

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Step 1.1.5.11.2.1

Move $x$ .

$f' (x) = 3 \cdot (4 (x \cdot x^{2})) + 3 x^{2} \cdot - 3 - 18 x (4 x) - 18 x \cdot - 3 + 27 (4 x) + 27 \cdot - 3$

Step 1.1.5.11.2.2

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

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Step 1.1.5.11.2.2.1

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

$f' (x) = 3 \cdot (4 (x \cdot x^{2})) + 3 x^{2} \cdot - 3 - 18 x (4 x) - 18 x \cdot - 3 + 27 (4 x) + 27 \cdot - 3$

Step 1.1.5.11.2.2.2

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

$f' (x) = 3 \cdot (4 x^{1 + 2}) + 3 x^{2} \cdot - 3 - 18 x (4 x) - 18 x \cdot - 3 + 27 (4 x) + 27 \cdot - 3$

Step 1.1.5.11.2.3

Add $1$ and $2$ .

$f' (x) = 3 \cdot (4 x^{3}) + 3 x^{2} \cdot - 3 - 18 x (4 x) - 18 x \cdot - 3 + 27 (4 x) + 27 \cdot - 3$

Step 1.1.5.11.3

Multiply $3$ by $4$ .

$f' (x) = 12 x^{3} + 3 x^{2} \cdot - 3 - 18 x (4 x) - 18 x \cdot - 3 + 27 (4 x) + 27 \cdot - 3$

Step 1.1.5.11.4

Multiply $- 3$ by $3$ .

$f' (x) = 12 x^{3} - 9 x^{2} - 18 x (4 x) - 18 x \cdot - 3 + 27 (4 x) + 27 \cdot - 3$

Step 1.1.5.11.5

Rewrite using the commutative property of multiplication.

$f' (x) = 12 x^{3} - 9 x^{2} - 18 \cdot (4 x \cdot x) - 18 x \cdot - 3 + 27 (4 x) + 27 \cdot - 3$

Step 1.1.5.11.6

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

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Step 1.1.5.11.6.1

Move $x$ .

$f' (x) = 12 x^{3} - 9 x^{2} - 18 \cdot (4 (x \cdot x)) - 18 x \cdot - 3 + 27 (4 x) + 27 \cdot - 3$

Step 1.1.5.11.6.2

Multiply $x$ by $x$ .

$f' (x) = 12 x^{3} - 9 x^{2} - 18 \cdot (4 x^{2}) - 18 x \cdot - 3 + 27 (4 x) + 27 \cdot - 3$

Step 1.1.5.11.7

Multiply $- 18$ by $4$ .

$f' (x) = 12 x^{3} - 9 x^{2} - 72 x^{2} - 18 x \cdot - 3 + 27 (4 x) + 27 \cdot - 3$

Step 1.1.5.11.8

Multiply $- 3$ by $- 18$ .

$f' (x) = 12 x^{3} - 9 x^{2} - 72 x^{2} + 54 x + 27 (4 x) + 27 \cdot - 3$

Step 1.1.5.11.9

Multiply $4$ by $27$ .

$f' (x) = 12 x^{3} - 9 x^{2} - 72 x^{2} + 54 x + 108 x + 27 \cdot - 3$

Step 1.1.5.11.10

Multiply $27$ by $- 3$ .

$f' (x) = 12 x^{3} - 9 x^{2} - 72 x^{2} + 54 x + 108 x - 81$

Step 1.1.5.12

Subtract $72 x^{2}$ from $- 9 x^{2}$ .

$f' (x) = 12 x^{3} - 81 x^{2} + 54 x + 108 x - 81$

Step 1.1.5.13

Add $54 x$ and $108 x$ .

$f' (x) = 12 x^{3} - 81 x^{2} + 162 x - 81$

Step 1.2

Find the second derivative.

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Step 1.2.1

By the Sum Rule, the derivative of $12 x^{3} - 81 x^{2} + 162 x - 81$ with respect to $x$ is $\frac{d}{d x} [12 x^{3}] + \frac{d}{d x} [- 81 x^{2}] + \frac{d}{d x} [162 x] + \frac{d}{d x} [- 81]$ .

$f'' (x) = \frac{d}{d x} (12 x^{3}) + \frac{d}{d x} (- 81 x^{2}) + \frac{d}{d x} (162 x) + \frac{d}{d x} (- 81)$

Step 1.2.2

Evaluate $\frac{d}{d x} [12 x^{3}]$ .

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Step 1.2.2.1

Since $12$ is constant with respect to $x$ , the derivative of $12 x^{3}$ with respect to $x$ is $12 \frac{d}{d x} [x^{3}]$ .

$f'' (x) = 12 \frac{d}{d x} (x^{3}) + \frac{d}{d x} (- 81 x^{2}) + \frac{d}{d x} (162 x) + \frac{d}{d x} (- 81)$

Step 1.2.2.2

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

$f'' (x) = 12 (3 x^{2}) + \frac{d}{d x} (- 81 x^{2}) + \frac{d}{d x} (162 x) + \frac{d}{d x} (- 81)$

Step 1.2.2.3

Multiply $3$ by $12$ .

$f'' (x) = 36 x^{2} + \frac{d}{d x} (- 81 x^{2}) + \frac{d}{d x} (162 x) + \frac{d}{d x} (- 81)$

Step 1.2.3

Evaluate $\frac{d}{d x} [- 81 x^{2}]$ .

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Step 1.2.3.1

Since $- 81$ is constant with respect to $x$ , the derivative of $- 81 x^{2}$ with respect to $x$ is $- 81 \frac{d}{d x} [x^{2}]$ .

$f'' (x) = 36 x^{2} - 81 \frac{d}{d x} x^{2} + \frac{d}{d x} (162 x) + \frac{d}{d x} (- 81)$

Step 1.2.3.2

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

$f'' (x) = 36 x^{2} - 81 (2 x) + \frac{d}{d x} (162 x) + \frac{d}{d x} (- 81)$

Step 1.2.3.3

Multiply $2$ by $- 81$ .

$f'' (x) = 36 x^{2} - 162 x + \frac{d}{d x} (162 x) + \frac{d}{d x} (- 81)$

Step 1.2.4

Evaluate $\frac{d}{d x} [162 x]$ .

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Step 1.2.4.1

Since $162$ is constant with respect to $x$ , the derivative of $162 x$ with respect to $x$ is $162 \frac{d}{d x} [x]$ .

$f'' (x) = 36 x^{2} - 162 x + 162 \frac{d}{d x} (x) + \frac{d}{d x} (- 81)$

Step 1.2.4.2

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

$f'' (x) = 36 x^{2} - 162 x + 162 \cdot 1 + \frac{d}{d x} (- 81)$

Step 1.2.4.3

Multiply $162$ by $1$ .

$f'' (x) = 36 x^{2} - 162 x + 162 + \frac{d}{d x} (- 81)$

Step 1.2.5

Differentiate using the Constant Rule.

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Step 1.2.5.1

Since $- 81$ is constant with respect to $x$ , the derivative of $- 81$ with respect to $x$ is $0$ .

$f'' (x) = 36 x^{2} - 162 x + 162 + 0$

Step 1.2.5.2

Add $36 x^{2} - 162 x + 162$ and $0$ .

$f'' (x) = 36 x^{2} - 162 x + 162$

Step 1.3

The second derivative of $f (x)$ with respect to $x$ is $36 x^{2} - 162 x + 162$ .

$36 x^{2} - 162 x + 162$

Step 2

Set the second derivative equal to $0$ then solve the equation $36 x^{2} - 162 x + 162 = 0$ .

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Step 2.1

Set the second derivative equal to $0$ .

$36 x^{2} - 162 x + 162 = 0$

Step 2.2

Factor the left side of the equation.

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Step 2.2.1

Factor $18$ out of $36 x^{2} - 162 x + 162$ .

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Step 2.2.1.1

Factor $18$ out of $36 x^{2}$ .

$18 (2 x^{2}) - 162 x + 162 = 0$

Step 2.2.1.2

Factor $18$ out of $- 162 x$ .

$18 (2 x^{2}) + 18 (- 9 x) + 162 = 0$

Step 2.2.1.3

Factor $18$ out of $162$ .

$18 (2 x^{2}) + 18 (- 9 x) + 18 (9) = 0$

Step 2.2.1.4

Factor $18$ out of $18 (2 x^{2}) + 18 (- 9 x)$ .

$18 (2 x^{2} - 9 x) + 18 (9) = 0$

Step 2.2.1.5

Factor $18$ out of $18 (2 x^{2} - 9 x) + 18 (9)$ .

$18 (2 x^{2} - 9 x + 9) = 0$

Step 2.2.2

Factor.

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Step 2.2.2.1

Factor by grouping.

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Step 2.2.2.1.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 2 \cdot 9 = 18$ and whose sum is $b = - 9$ .

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Step 2.2.2.1.1.1

Factor $- 9$ out of $- 9 x$ .

$18 (2 x^{2} - 9 x + 9) = 0$

Step 2.2.2.1.1.2

Rewrite $- 9$ as $- 3$ plus $- 6$

$18 (2 x^{2} + (- 3 - 6) x + 9) = 0$

Step 2.2.2.1.1.3

Apply the distributive property.

$18 (2 x^{2} - 3 x - 6 x + 9) = 0$

Step 2.2.2.1.2

Factor out the greatest common factor from each group.

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Step 2.2.2.1.2.1

Group the first two terms and the last two terms.

$18 ((2 x^{2} - 3 x) - 6 x + 9) = 0$

Step 2.2.2.1.2.2

Factor out the greatest common factor (GCF) from each group.

$18 (x (2 x - 3) - 3 (2 x - 3)) = 0$

Step 2.2.2.1.3

Factor the polynomial by factoring out the greatest common factor, $2 x - 3$ .

$18 ((2 x - 3) (x - 3)) = 0$

Step 2.2.2.2

Remove unnecessary parentheses.

$18 (2 x - 3) (x - 3) = 0$

Step 2.3

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$2 x - 3 = 0$

$x - 3 = 0$

Step 2.4

Set $2 x - 3$ equal to $0$ and solve for $x$ .

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Step 2.4.1

Set $2 x - 3$ equal to $0$ .

$2 x - 3 = 0$

Step 2.4.2

Solve $2 x - 3 = 0$ for $x$ .

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Step 2.4.2.1

Add $3$ to both sides of the equation.

$2 x = 3$

Step 2.4.2.2

Divide each term in $2 x = 3$ by $2$ and simplify.

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Step 2.4.2.2.1

Divide each term in $2 x = 3$ by $2$ .

$\frac{2 x}{2} = \frac{3}{2}$

Step 2.4.2.2.2

Simplify the left side.

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Step 2.4.2.2.2.1

Cancel the common factor of $2$ .

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Step 2.4.2.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{3}{2}$

Step 2.4.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{3}{2}$

Step 2.5

Set $x - 3$ equal to $0$ and solve for $x$ .

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Step 2.5.1

Set $x - 3$ equal to $0$ .

$x - 3 = 0$

Step 2.5.2

Add $3$ to both sides of the equation.

$x = 3$

Step 2.6

The final solution is all the values that make $18 (2 x - 3) (x - 3) = 0$ true.

$x = \frac{3}{2}, 3$

Step 3

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

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Step 3.1

Substitute $\frac{3}{2}$ in $f (x) = 3 x {(x - 3)}^{3}$ to find the value of $y$ .

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Step 3.1.1

Replace the variable $x$ with $\frac{3}{2}$ in the expression.

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

Step 3.1.2

Simplify the result.

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Step 3.1.2.1

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

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Step 3.1.2.1.1

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

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

Step 3.1.2.1.2

Multiply $3$ by $3$ .

$f (\frac{3}{2}) = \frac{9}{2} \cdot {((\frac{3}{2}) - 3)}^{3}$

Step 3.1.2.2

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

$f (\frac{3}{2}) = \frac{9}{2} \cdot {(\frac{3}{2} - 3 \cdot \frac{2}{2})}^{3}$

Step 3.1.2.3

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

$f (\frac{3}{2}) = \frac{9}{2} \cdot {(\frac{3}{2} + \frac{- 3 \cdot 2}{2})}^{3}$

Step 3.1.2.4

Combine the numerators over the common denominator.

$f (\frac{3}{2}) = \frac{9}{2} \cdot {(\frac{3 - 3 \cdot 2}{2})}^{3}$

Step 3.1.2.5

Simplify the numerator.

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Step 3.1.2.5.1

Multiply $- 3$ by $2$ .

$f (\frac{3}{2}) = \frac{9}{2} \cdot {(\frac{3 - 6}{2})}^{3}$

Step 3.1.2.5.2

Subtract $6$ from $3$ .

$f (\frac{3}{2}) = \frac{9}{2} \cdot {(\frac{- 3}{2})}^{3}$

Step 3.1.2.6

Move the negative in front of the fraction.

$f (\frac{3}{2}) = \frac{9}{2} \cdot {(- \frac{3}{2})}^{3}$

Step 3.1.2.7

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

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Step 3.1.2.7.1

Apply the product rule to $- \frac{3}{2}$ .

$f (\frac{3}{2}) = \frac{9}{2} \cdot ({(- 1)}^{3} {(\frac{3}{2})}^{3})$

Step 3.1.2.7.2

Apply the product rule to $\frac{3}{2}$ .

$f (\frac{3}{2}) = \frac{9}{2} \cdot ({(- 1)}^{3} (\frac{3^{3}}{2^{3}}))$

Step 3.1.2.8

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

$f (\frac{3}{2}) = \frac{9}{2} \cdot (- \frac{3^{3}}{2^{3}})$

Step 3.1.2.9

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

$f (\frac{3}{2}) = \frac{9}{2} \cdot (- \frac{27}{2^{3}})$

Step 3.1.2.10

Raise $2$ to the power of $3$ .

$f (\frac{3}{2}) = \frac{9}{2} \cdot (- \frac{27}{8})$

Step 3.1.2.11

Multiply $\frac{9}{2} (- \frac{27}{8})$ .

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Step 3.1.2.11.1

Multiply $\frac{9}{2}$ by $\frac{27}{8}$ .

$f (\frac{3}{2}) = - \frac{9 \cdot 27}{2 \cdot 8}$

Step 3.1.2.11.2

Multiply $9$ by $27$ .

$f (\frac{3}{2}) = - \frac{243}{2 \cdot 8}$

Step 3.1.2.11.3

Multiply $2$ by $8$ .

$f (\frac{3}{2}) = - \frac{243}{16}$

Step 3.1.2.12

The final answer is $- \frac{243}{16}$ .

$- \frac{243}{16}$

Step 3.2

The point found by substituting $\frac{3}{2}$ in $f (x) = 3 x {(x - 3)}^{3}$ is $(\frac{3}{2}, - \frac{243}{16})$ . This point can be an inflection point.

$(\frac{3}{2}, - \frac{243}{16})$

Step 3.3

Substitute $3$ in $f (x) = 3 x {(x - 3)}^{3}$ to find the value of $y$ .

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Step 3.3.1

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

$f (3) = 3 (3) {((3) - 3)}^{3}$

Step 3.3.2

Simplify the result.

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Step 3.3.2.1

Multiply $3$ by $3$ .

$f (3) = 9 {((3) - 3)}^{3}$

Step 3.3.2.2

Subtract $3$ from $3$ .

$f (3) = 9 \cdot 0^{3}$

Step 3.3.2.3

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

$f (3) = 9 \cdot 0$

Step 3.3.2.4

Multiply $9$ by $0$ .

$f (3) = 0$

Step 3.3.2.5

The final answer is $0$ .

$0$

Step 3.4

The point found by substituting $3$ in $f (x) = 3 x {(x - 3)}^{3}$ is $(3, 0)$ . This point can be an inflection point.

$(3, 0)$

Step 3.5

Determine the points that could be inflection points.

$(\frac{3}{2}, - \frac{243}{16}), (3, 0)$

Step 4

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

$(- \infty, \frac{3}{2}) \cup (\frac{3}{2}, 3) \cup (3, \infty)$

Step 5

Substitute a value from the interval $(- \infty, \frac{3}{2})$ into the second derivative to determine if it is increasing or decreasing.

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Step 5.1

Replace the variable $x$ with $1.4$ in the expression.

$f'' (1.4) = 36 {(1.4)}^{2} - 162 \cdot 1.4 + 162$

Step 5.2

Simplify the result.

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Step 5.2.1

Simplify each term.

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Step 5.2.1.1

Raise $1.4$ to the power of $2$ .

$f'' (1.4) = 36 \cdot 1.96 - 162 \cdot 1.4 + 162$

Step 5.2.1.2

Multiply $36$ by $1.96$ .

$f'' (1.4) = 70.56 - 162 \cdot 1.4 + 162$

Step 5.2.1.3

Multiply $- 162$ by $1.4$ .

$f'' (1.4) = 70.56 - 226.8 + 162$

Step 5.2.2

Simplify by adding and subtracting.

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Step 5.2.2.1

Subtract $226.8$ from $70.56$ .

$f'' (1.4) = - 156.24 + 162$

Step 5.2.2.2

Add $- 156.24$ and $162$ .

$f'' (1.4) = 5.76$

Step 5.2.3

The final answer is $5.76$ .

$5.76$

Step 5.3

At $1.4$ , the second derivative is $5.76$ . Since this is positive, the second derivative is increasing on the interval $(- \infty, \frac{3}{2})$ .

Increasing on $(- \infty, \frac{3}{2})$ since $f'' (x) > 0$

Step 6

Substitute a value from the interval $(\frac{3}{2}, 3)$ into the second derivative to determine if it is increasing or decreasing.

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Step 6.1

Replace the variable $x$ with $2.25$ in the expression.

$f'' (2.25) = 36 {(2.25)}^{2} - 162 \cdot 2.25 + 162$

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 $2.25$ to the power of $2$ .

$f'' (2.25) = 36 \cdot 5.0625 - 162 \cdot 2.25 + 162$

Step 6.2.1.2

Multiply $36$ by $5.0625$ .

$f'' (2.25) = 182.25 - 162 \cdot 2.25 + 162$

Step 6.2.1.3

Multiply $- 162$ by $2.25$ .

$f'' (2.25) = 182.25 - 364.5 + 162$

Step 6.2.2

Simplify by adding and subtracting.

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Step 6.2.2.1

Subtract $364.5$ from $182.25$ .

$f'' (2.25) = - 182.25 + 162$

Step 6.2.2.2

Add $- 182.25$ and $162$ .

$f'' (2.25) = - 20.25$

Step 6.2.3

The final answer is $- 20.25$ .

$- 20.25$

Step 6.3

At $2.25$ , the second derivative is $- 20.25$ . Since this is negative, the second derivative is decreasing on the interval $(\frac{3}{2}, 3)$

Decreasing on $(\frac{3}{2}, 3)$ since $f'' (x) < 0$

Step 7

Substitute a value from the interval $(3, \infty)$ into the second derivative to determine if it is increasing or decreasing.

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Step 7.1

Replace the variable $x$ with $3.1$ in the expression.

$f'' (3.1) = 36 {(3.1)}^{2} - 162 \cdot 3.1 + 162$

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

Raise $3.1$ to the power of $2$ .

$f'' (3.1) = 36 \cdot 9.61 - 162 \cdot 3.1 + 162$

Step 7.2.1.2

Multiply $36$ by $9.61$ .

$f'' (3.1) = 345.96 - 162 \cdot 3.1 + 162$

Step 7.2.1.3

Multiply $- 162$ by $3.1$ .

$f'' (3.1) = 345.96 - 502.2 + 162$

Step 7.2.2

Simplify by adding and subtracting.

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Step 7.2.2.1

Subtract $502.2$ from $345.96$ .

$f'' (3.1) = - 156.24 + 162$

Step 7.2.2.2

Add $- 156.24$ and $162$ .

$f'' (3.1) = 5.76$

Step 7.2.3

The final answer is $5.76$ .

$5.76$

Step 7.3

At $3.1$ , the second derivative is $5.76$ . Since this is positive, the second derivative is increasing on the interval $(3, \infty)$ .

Increasing on $(3, \infty)$ since $f'' (x) > 0$

Step 8

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 $(\frac{3}{2}, - \frac{243}{16}), (3, 0)$ .

$(\frac{3}{2}, - \frac{243}{16}), (3, 0)$

Step 9