Calculus Examples

$f (x) = 3 x^{4} + 16 x^{3} + 24 x^{2} + 32$

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

By the Sum Rule, the derivative of $3 x^{4} + 16 x^{3} + 24 x^{2} + 32$ with respect to $x$ is $\frac{d}{d x} [3 x^{4}] + \frac{d}{d x} [16 x^{3}] + \frac{d}{d x} [24 x^{2}] + \frac{d}{d x} [32]$ .

$f' (x) = \frac{d}{d x} (3 x^{4}) + \frac{d}{d x} (16 x^{3}) + \frac{d}{d x} (24 x^{2}) + \frac{d}{d x} (32)$

Step 1.1.2

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

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Step 1.1.2.1

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

$f' (x) = 3 \frac{d}{d x} (x^{4}) + \frac{d}{d x} (16 x^{3}) + \frac{d}{d x} (24 x^{2}) + \frac{d}{d x} (32)$

Step 1.1.2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 4$ .

$f' (x) = 3 (4 x^{3}) + \frac{d}{d x} (16 x^{3}) + \frac{d}{d x} (24 x^{2}) + \frac{d}{d x} (32)$

Step 1.1.2.3

Multiply $4$ by $3$ .

$f' (x) = 12 x^{3} + \frac{d}{d x} (16 x^{3}) + \frac{d}{d x} (24 x^{2}) + \frac{d}{d x} (32)$

Step 1.1.3

Evaluate $\frac{d}{d x} [16 x^{3}]$ .

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Step 1.1.3.1

Since $16$ is constant with respect to $x$ , the derivative of $16 x^{3}$ with respect to $x$ is $16 \frac{d}{d x} [x^{3}]$ .

$f' (x) = 12 x^{3} + 16 \frac{d}{d x} (x^{3}) + \frac{d}{d x} (24 x^{2}) + \frac{d}{d x} (32)$

Step 1.1.3.2

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

$f' (x) = 12 x^{3} + 16 (3 x^{2}) + \frac{d}{d x} (24 x^{2}) + \frac{d}{d x} (32)$

Step 1.1.3.3

Multiply $3$ by $16$ .

$f' (x) = 12 x^{3} + 48 x^{2} + \frac{d}{d x} (24 x^{2}) + \frac{d}{d x} (32)$

Step 1.1.4

Evaluate $\frac{d}{d x} [24 x^{2}]$ .

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Step 1.1.4.1

Since $24$ is constant with respect to $x$ , the derivative of $24 x^{2}$ with respect to $x$ is $24 \frac{d}{d x} [x^{2}]$ .

$f' (x) = 12 x^{3} + 48 x^{2} + 24 \frac{d}{d x} (x^{2}) + \frac{d}{d x} (32)$

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 $n = 2$ .

$f' (x) = 12 x^{3} + 48 x^{2} + 24 (2 x) + \frac{d}{d x} (32)$

Step 1.1.4.3

Multiply $2$ by $24$ .

$f' (x) = 12 x^{3} + 48 x^{2} + 48 x + \frac{d}{d x} (32)$

Step 1.1.5

Differentiate using the Constant Rule.

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Step 1.1.5.1

Since $32$ is constant with respect to $x$ , the derivative of $32$ with respect to $x$ is $0$ .

$f' (x) = 12 x^{3} + 48 x^{2} + 48 x + 0$

Step 1.1.5.2

Add $12 x^{3} + 48 x^{2} + 48 x$ and $0$ .

$f' (x) = 12 x^{3} + 48 x^{2} + 48 x$

Step 1.2

Find the second derivative.

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Step 1.2.1

By the Sum Rule, the derivative of $12 x^{3} + 48 x^{2} + 48 x$ with respect to $x$ is $\frac{d}{d x} [12 x^{3}] + \frac{d}{d x} [48 x^{2}] + \frac{d}{d x} [48 x]$ .

$f'' (x) = \frac{d}{d x} (12 x^{3}) + \frac{d}{d x} (48 x^{2}) + \frac{d}{d x} (48 x)$

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

$f'' (x) = 12 \frac{d}{d x} (x^{3}) + \frac{d}{d x} (48 x^{2}) + \frac{d}{d x} (48 x)$

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 $n = 3$ .

$f'' (x) = 12 (3 x^{2}) + \frac{d}{d x} (48 x^{2}) + \frac{d}{d x} (48 x)$

Step 1.2.2.3

Multiply $3$ by $12$ .

$f'' (x) = 36 x^{2} + \frac{d}{d x} (48 x^{2}) + \frac{d}{d x} (48 x)$

Step 1.2.3

Evaluate $\frac{d}{d x} [48 x^{2}]$ .

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Step 1.2.3.1

Since $48$ is constant with respect to $x$ , the derivative of $48 x^{2}$ with respect to $x$ is $48 \frac{d}{d x} [x^{2}]$ .

$f'' (x) = 36 x^{2} + 48 \frac{d}{d x} (x^{2}) + \frac{d}{d x} (48 x)$

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 $n = 2$ .

$f'' (x) = 36 x^{2} + 48 (2 x) + \frac{d}{d x} (48 x)$

Step 1.2.3.3

Multiply $2$ by $48$ .

$f'' (x) = 36 x^{2} + 96 x + \frac{d}{d x} (48 x)$

Step 1.2.4

Evaluate $\frac{d}{d x} [48 x]$ .

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Step 1.2.4.1

Since $48$ is constant with respect to $x$ , the derivative of $48 x$ with respect to $x$ is $48 \frac{d}{d x} [x]$ .

$f'' (x) = 36 x^{2} + 96 x + 48 \frac{d}{d x} (x)$

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 $n = 1$ .

$f'' (x) = 36 x^{2} + 96 x + 48 \cdot 1$

Step 1.2.4.3

Multiply $48$ by $1$ .

$f'' (x) = 36 x^{2} + 96 x + 48$

Step 1.3

The second derivative of $f (x)$ with respect to $x$ is $36 x^{2} + 96 x + 48$ .

$36 x^{2} + 96 x + 48$

Step 2

Set the second derivative equal to $0$ then solve the equation $36 x^{2} + 96 x + 48 = 0$ .

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Step 2.1

Set the second derivative equal to $0$ .

$36 x^{2} + 96 x + 48 = 0$

Step 2.2

Factor the left side of the equation.

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Step 2.2.1

Factor $12$ out of $36 x^{2} + 96 x + 48$ .

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Step 2.2.1.1

Factor $12$ out of $36 x^{2}$ .

$12 (3 x^{2}) + 96 x + 48 = 0$

Step 2.2.1.2

Factor $12$ out of $96 x$ .

$12 (3 x^{2}) + 12 (8 x) + 48 = 0$

Step 2.2.1.3

Factor $12$ out of $48$ .

$12 (3 x^{2}) + 12 (8 x) + 12 (4) = 0$

Step 2.2.1.4

Factor $12$ out of $12 (3 x^{2}) + 12 (8 x)$ .

$12 (3 x^{2} + 8 x) + 12 (4) = 0$

Step 2.2.1.5

Factor $12$ out of $12 (3 x^{2} + 8 x) + 12 (4)$ .

$12 (3 x^{2} + 8 x + 4) = 0$

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 $a \cdot c = 3 \cdot 4 = 12$ and whose sum is $b = 8$ .

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Step 2.2.2.1.1.1

Factor $8$ out of $8 x$ .

$12 (3 x^{2} + 8 (x) + 4) = 0$

Step 2.2.2.1.1.2

Rewrite $8$ as $2$ plus $6$

$12 (3 x^{2} + (2 + 6) x + 4) = 0$

Step 2.2.2.1.1.3

Apply the distributive property.

$12 (3 x^{2} + 2 x + 6 x + 4) = 0$

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.

$12 ((3 x^{2} + 2 x) + 6 x + 4) = 0$

Step 2.2.2.1.2.2

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

$12 (x (3 x + 2) + 2 (3 x + 2)) = 0$

Step 2.2.2.1.3

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

$12 ((3 x + 2) (x + 2)) = 0$

Step 2.2.2.2

Remove unnecessary parentheses.

$12 (3 x + 2) (x + 2) = 0$

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 $0$ .

$3 x + 2 = 0$

$x + 2 = 0$

Step 2.4

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

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Step 2.4.1

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

$3 x + 2 = 0$

Step 2.4.2

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

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Step 2.4.2.1

Subtract $2$ from both sides of the equation.

$3 x = - 2$

Step 2.4.2.2

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

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Step 2.4.2.2.1

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

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

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 $3$ .

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Step 2.4.2.2.2.1.1

Cancel the common factor.

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

Step 2.4.2.2.2.1.2

Divide $x$ by $1$ .

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

Step 2.4.2.2.3

Simplify the right side.

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Step 2.4.2.2.3.1

Move the negative in front of the fraction.

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

Step 2.5

Set $x + 2$ equal to $0$ and solve for $x$ .

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Step 2.5.1

Set $x + 2$ equal to $0$ .

$x + 2 = 0$

Step 2.5.2

Subtract $2$ from both sides of the equation.

$x = - 2$

Step 2.6

The final solution is all the values that make $12 (3 x + 2) (x + 2) = 0$ true.

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

Step 3

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

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Step 3.1

Substitute $- \frac{2}{3}$ in $f (x) = 3 x^{4} + 16 x^{3} + 24 x^{2} + 32$ to find the value of $y$ .

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Step 3.1.1

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

$f (- \frac{2}{3}) = 3 {(- \frac{2}{3})}^{4} + 16 {(- \frac{2}{3})}^{3} + 24 {(- \frac{2}{3})}^{2} + 32$

Step 3.1.2

Simplify the result.

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Step 3.1.2.1

Simplify each term.

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Step 3.1.2.1.1

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

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Step 3.1.2.1.1.1

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

$f (- \frac{2}{3}) = 3 ({(- 1)}^{4} {(\frac{2}{3})}^{4}) + 16 {(- \frac{2}{3})}^{3} + 24 {(- \frac{2}{3})}^{2} + 32$

Step 3.1.2.1.1.2

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

$f (- \frac{2}{3}) = 3 ({(- 1)}^{4} (\frac{2^{4}}{3^{4}})) + 16 {(- \frac{2}{3})}^{3} + 24 {(- \frac{2}{3})}^{2} + 32$

Step 3.1.2.1.2

Raise $- 1$ to the power of $4$ .

$f (- \frac{2}{3}) = 3 (1 (\frac{2^{4}}{3^{4}})) + 16 {(- \frac{2}{3})}^{3} + 24 {(- \frac{2}{3})}^{2} + 32$

Step 3.1.2.1.3

Multiply $\frac{2^{4}}{3^{4}}$ by $1$ .

$f (- \frac{2}{3}) = 3 (\frac{2^{4}}{3^{4}}) + 16 {(- \frac{2}{3})}^{3} + 24 {(- \frac{2}{3})}^{2} + 32$

Step 3.1.2.1.4

Raise $2$ to the power of $4$ .

$f (- \frac{2}{3}) = 3 (\frac{16}{3^{4}}) + 16 {(- \frac{2}{3})}^{3} + 24 {(- \frac{2}{3})}^{2} + 32$

Step 3.1.2.1.5

Raise $3$ to the power of $4$ .

$f (- \frac{2}{3}) = 3 (\frac{16}{81}) + 16 {(- \frac{2}{3})}^{3} + 24 {(- \frac{2}{3})}^{2} + 32$

Step 3.1.2.1.6

Cancel the common factor of $3$ .

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Step 3.1.2.1.6.1

Factor $3$ out of $81$ .

$f (- \frac{2}{3}) = 3 (\frac{16}{3 (27)}) + 16 {(- \frac{2}{3})}^{3} + 24 {(- \frac{2}{3})}^{2} + 32$

Step 3.1.2.1.6.2

Cancel the common factor.

$f (- \frac{2}{3}) = 3 (\frac{16}{3 \cdot 27}) + 16 {(- \frac{2}{3})}^{3} + 24 {(- \frac{2}{3})}^{2} + 32$

Step 3.1.2.1.6.3

Rewrite the expression.

$f (- \frac{2}{3}) = \frac{16}{27} + 16 {(- \frac{2}{3})}^{3} + 24 {(- \frac{2}{3})}^{2} + 32$

Step 3.1.2.1.7

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

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Step 3.1.2.1.7.1

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

$f (- \frac{2}{3}) = \frac{16}{27} + 16 ({(- 1)}^{3} {(\frac{2}{3})}^{3}) + 24 {(- \frac{2}{3})}^{2} + 32$

Step 3.1.2.1.7.2

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

$f (- \frac{2}{3}) = \frac{16}{27} + 16 ({(- 1)}^{3} (\frac{2^{3}}{3^{3}})) + 24 {(- \frac{2}{3})}^{2} + 32$

Step 3.1.2.1.8

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

$f (- \frac{2}{3}) = \frac{16}{27} + 16 (- \frac{2^{3}}{3^{3}}) + 24 {(- \frac{2}{3})}^{2} + 32$

Step 3.1.2.1.9

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

$f (- \frac{2}{3}) = \frac{16}{27} + 16 (- \frac{8}{3^{3}}) + 24 {(- \frac{2}{3})}^{2} + 32$

Step 3.1.2.1.10

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

$f (- \frac{2}{3}) = \frac{16}{27} + 16 (- \frac{8}{27}) + 24 {(- \frac{2}{3})}^{2} + 32$

Step 3.1.2.1.11

Multiply $16 (- \frac{8}{27})$ .

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Step 3.1.2.1.11.1

Multiply $- 1$ by $16$ .

$f (- \frac{2}{3}) = \frac{16}{27} - 16 (\frac{8}{27}) + 24 {(- \frac{2}{3})}^{2} + 32$

Step 3.1.2.1.11.2

Combine $- 16$ and $\frac{8}{27}$ .

$f (- \frac{2}{3}) = \frac{16}{27} + \frac{- 16 \cdot 8}{27} + 24 {(- \frac{2}{3})}^{2} + 32$

Step 3.1.2.1.11.3

Multiply $- 16$ by $8$ .

$f (- \frac{2}{3}) = \frac{16}{27} + \frac{- 128}{27} + 24 {(- \frac{2}{3})}^{2} + 32$

Step 3.1.2.1.12

Move the negative in front of the fraction.

$f (- \frac{2}{3}) = \frac{16}{27} - \frac{128}{27} + 24 {(- \frac{2}{3})}^{2} + 32$

Step 3.1.2.1.13

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

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Step 3.1.2.1.13.1

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

$f (- \frac{2}{3}) = \frac{16}{27} - \frac{128}{27} + 24 ({(- 1)}^{2} {(\frac{2}{3})}^{2}) + 32$

Step 3.1.2.1.13.2

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

$f (- \frac{2}{3}) = \frac{16}{27} - \frac{128}{27} + 24 ({(- 1)}^{2} (\frac{2^{2}}{3^{2}})) + 32$

Step 3.1.2.1.14

Raise $- 1$ to the power of $2$ .

$f (- \frac{2}{3}) = \frac{16}{27} - \frac{128}{27} + 24 (1 (\frac{2^{2}}{3^{2}})) + 32$

Step 3.1.2.1.15

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

$f (- \frac{2}{3}) = \frac{16}{27} - \frac{128}{27} + 24 (\frac{2^{2}}{3^{2}}) + 32$

Step 3.1.2.1.16

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

$f (- \frac{2}{3}) = \frac{16}{27} - \frac{128}{27} + 24 (\frac{4}{3^{2}}) + 32$

Step 3.1.2.1.17

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

$f (- \frac{2}{3}) = \frac{16}{27} - \frac{128}{27} + 24 (\frac{4}{9}) + 32$

Step 3.1.2.1.18

Cancel the common factor of $3$ .

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Step 3.1.2.1.18.1

Factor $3$ out of $24$ .

$f (- \frac{2}{3}) = \frac{16}{27} - \frac{128}{27} + 3 (8) (\frac{4}{9}) + 32$

Step 3.1.2.1.18.2

Factor $3$ out of $9$ .

$f (- \frac{2}{3}) = \frac{16}{27} - \frac{128}{27} + 3 \cdot (8 (\frac{4}{3 \cdot 3})) + 32$

Step 3.1.2.1.18.3

Cancel the common factor.

$f (- \frac{2}{3}) = \frac{16}{27} - \frac{128}{27} + 3 \cdot (8 (\frac{4}{3 \cdot 3})) + 32$

Step 3.1.2.1.18.4

Rewrite the expression.

$f (- \frac{2}{3}) = \frac{16}{27} - \frac{128}{27} + 8 (\frac{4}{3}) + 32$

Step 3.1.2.1.19

Combine $8$ and $\frac{4}{3}$ .

$f (- \frac{2}{3}) = \frac{16}{27} - \frac{128}{27} + \frac{8 \cdot 4}{3} + 32$

Step 3.1.2.1.20

Multiply $8$ by $4$ .

$f (- \frac{2}{3}) = \frac{16}{27} - \frac{128}{27} + \frac{32}{3} + 32$

Step 3.1.2.2

Combine fractions.

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Step 3.1.2.2.1

Combine the numerators over the common denominator.

$f (- \frac{2}{3}) = 32 + \frac{16 - 128}{27} + \frac{32}{3}$

Step 3.1.2.2.2

Subtract $128$ from $16$ .

$f (- \frac{2}{3}) = 32 + \frac{- 112}{27} + \frac{32}{3}$

Step 3.1.2.3

Find the common denominator.

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Step 3.1.2.3.1

Write $32$ as a fraction with denominator $1$ .

$f (- \frac{2}{3}) = \frac{32}{1} + \frac{- 112}{27} + \frac{32}{3}$

Step 3.1.2.3.2

Multiply $\frac{32}{1}$ by $\frac{27}{27}$ .

$f (- \frac{2}{3}) = \frac{32}{1} \cdot \frac{27}{27} + \frac{- 112}{27} + \frac{32}{3}$

Step 3.1.2.3.3

Multiply $\frac{32}{1}$ by $\frac{27}{27}$ .

$f (- \frac{2}{3}) = \frac{32 \cdot 27}{27} + \frac{- 112}{27} + \frac{32}{3}$

Step 3.1.2.3.4

Multiply $\frac{32}{3}$ by $\frac{9}{9}$ .

$f (- \frac{2}{3}) = \frac{32 \cdot 27}{27} + \frac{- 112}{27} + \frac{32}{3} \cdot \frac{9}{9}$

Step 3.1.2.3.5

Multiply $\frac{32}{3}$ by $\frac{9}{9}$ .

$f (- \frac{2}{3}) = \frac{32 \cdot 27}{27} + \frac{- 112}{27} + \frac{32 \cdot 9}{3 \cdot 9}$

Step 3.1.2.3.6

Multiply $3$ by $9$ .

$f (- \frac{2}{3}) = \frac{32 \cdot 27}{27} + \frac{- 112}{27} + \frac{32 \cdot 9}{27}$

Step 3.1.2.4

Combine the numerators over the common denominator.

$f (- \frac{2}{3}) = \frac{32 \cdot 27 - 112 + 32 \cdot 9}{27}$

Step 3.1.2.5

Simplify each term.

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Step 3.1.2.5.1

Multiply $32$ by $27$ .

$f (- \frac{2}{3}) = \frac{864 - 112 + 32 \cdot 9}{27}$

Step 3.1.2.5.2

Multiply $32$ by $9$ .

$f (- \frac{2}{3}) = \frac{864 - 112 + 288}{27}$

Step 3.1.2.6

Simplify by adding and subtracting.

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Step 3.1.2.6.1

Subtract $112$ from $864$ .

$f (- \frac{2}{3}) = \frac{752 + 288}{27}$

Step 3.1.2.6.2

Add $752$ and $288$ .

$f (- \frac{2}{3}) = \frac{1040}{27}$

Step 3.1.2.7

The final answer is $\frac{1040}{27}$ .

$\frac{1040}{27}$

Step 3.2

The point found by substituting $- \frac{2}{3}$ in $f (x) = 3 x^{4} + 16 x^{3} + 24 x^{2} + 32$ is $(- \frac{2}{3}, \frac{1040}{27})$ . This point can be an inflection point.

$(- \frac{2}{3}, \frac{1040}{27})$

Step 3.3

Substitute $- 2$ in $f (x) = 3 x^{4} + 16 x^{3} + 24 x^{2} + 32$ to find the value of $y$ .

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Step 3.3.1

Replace the variable $x$ with $- 2$ in the expression.

$f (- 2) = 3 {(- 2)}^{4} + 16 {(- 2)}^{3} + 24 {(- 2)}^{2} + 32$

Step 3.3.2

Simplify the result.

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Step 3.3.2.1

Simplify each term.

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Step 3.3.2.1.1

Raise $- 2$ to the power of $4$ .

$f (- 2) = 3 \cdot 16 + 16 {(- 2)}^{3} + 24 {(- 2)}^{2} + 32$

Step 3.3.2.1.2

Multiply $3$ by $16$ .

$f (- 2) = 48 + 16 {(- 2)}^{3} + 24 {(- 2)}^{2} + 32$

Step 3.3.2.1.3

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

$f (- 2) = 48 + 16 \cdot - 8 + 24 {(- 2)}^{2} + 32$

Step 3.3.2.1.4

Multiply $16$ by $- 8$ .

$f (- 2) = 48 - 128 + 24 {(- 2)}^{2} + 32$

Step 3.3.2.1.5

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

$f (- 2) = 48 - 128 + 24 \cdot 4 + 32$

Step 3.3.2.1.6

Multiply $24$ by $4$ .

$f (- 2) = 48 - 128 + 96 + 32$

Step 3.3.2.2

Simplify by adding and subtracting.

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Step 3.3.2.2.1

Subtract $128$ from $48$ .

$f (- 2) = - 80 + 96 + 32$

Step 3.3.2.2.2

Add $- 80$ and $96$ .

$f (- 2) = 16 + 32$

Step 3.3.2.2.3

Add $16$ and $32$ .

$f (- 2) = 48$

Step 3.3.2.3

The final answer is $48$ .

$48$

Step 3.4

The point found by substituting $- 2$ in $f (x) = 3 x^{4} + 16 x^{3} + 24 x^{2} + 32$ is $(- 2, 48)$ . This point can be an inflection point.

$(- 2, 48)$

Step 3.5

Determine the points that could be inflection points.

$(- \frac{2}{3}, \frac{1040}{27}), (- 2, 48)$

Step 4

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

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

Step 5

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

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Step 5.1

Replace the variable $x$ with $- 2.1$ in the expression.

$f'' (- 2.1) = 36 {(- 2.1)}^{2} + 96 (- 2.1) + 48$

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 $- 2.1$ to the power of $2$ .

$f'' (- 2.1) = 36 \cdot 4.41 + 96 (- 2.1) + 48$

Step 5.2.1.2

Multiply $36$ by $4.41$ .

$f'' (- 2.1) = 158.76 + 96 (- 2.1) + 48$

Step 5.2.1.3

Multiply $96$ by $- 2.1$ .

$f'' (- 2.1) = 158.76 - 201.6 + 48$

Step 5.2.2

Simplify by adding and subtracting.

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Step 5.2.2.1

Subtract $201.6$ from $158.76$ .

$f'' (- 2.1) = - 42.84 + 48$

Step 5.2.2.2

Add $- 42.84$ and $48$ .

$f'' (- 2.1) = 5.16$

Step 5.2.3

The final answer is $5.16$ .

$5.16$

Step 5.3

At $- 2.1$ , the second derivative is $5.16$ . Since this is positive, the second derivative is increasing on the interval $(- \infty, - 2)$ .

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

Step 6

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

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Step 6.1

Replace the variable $x$ with $- 1.\overline{3}$ in the expression.

$f'' (- 1.\overline{3}) = 36 {(- 1.\overline{3})}^{2} + 96 (- 1.\overline{3}) + 48$

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 $- 1.\overline{3}$ to the power of $2$ .

$f'' (- 1.\overline{3}) = 36 \cdot 1.\overline{7} + 96 (- 1.\overline{3}) + 48$

Step 6.2.1.2

Multiply $36$ by $1.\overline{7}$ .

$f'' (- 1.\overline{3}) = 64 + 96 (- 1.\overline{3}) + 48$

Step 6.2.1.3

Multiply $96$ by $- 1.\overline{3}$ .

$f'' (- 1.\overline{3}) = 64 - 128 + 48$

Step 6.2.2

Simplify by adding and subtracting.

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Step 6.2.2.1

Subtract $128$ from $64$ .

$f'' (- 1.\overline{3}) = - 64 + 48$

Step 6.2.2.2

Add $- 64$ and $48$ .

$f'' (- 1.\overline{3}) = - 16$

Step 6.2.3

The final answer is $- 16$ .

$- 16$

Step 6.3

At $- 1.\overline{3}$ , the second derivative is $- 16$ . Since this is negative, the second derivative is decreasing on the interval $(- 2, - \frac{2}{3})$

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

Step 7

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

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Step 7.1

Replace the variable $x$ with $- 0.5\overline{6}$ in the expression.

$f'' (- 0.5\overline{6}) = 36 {(- 0.5\overline{6})}^{2} + 96 (- 0.5\overline{6}) + 48$

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 $- 0.5\overline{6}$ to the power of $2$ .

$f'' (- 0.5\overline{6}) = 36 \cdot 0.32\overline{1} + 96 (- 0.5\overline{6}) + 48$

Step 7.2.1.2

Multiply $36$ by $0.32\overline{1}$ .

$f'' (- 0.5\overline{6}) = 11.55\overline{9} + 96 (- 0.5\overline{6}) + 48$

Step 7.2.1.3

Multiply $96$ by $- 0.5\overline{6}$ .

$f'' (- 0.5\overline{6}) = 11.55\overline{9} - 54.4 + 48$

Step 7.2.2

Simplify by adding and subtracting.

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Step 7.2.2.1

Subtract $54.4$ from $11.55\overline{9}$ .

$f'' (- 0.5\overline{6}) = - 42.84 + 48$

Step 7.2.2.2

Add $- 42.84$ and $48$ .

$f'' (- 0.5\overline{6}) = 5.16$

Step 7.2.3

The final answer is $5.16$ .

$5.16$

Step 7.3

At $- 0.5\overline{6}$ , the second derivative is $5.16$ . Since this is positive, the second derivative is increasing on the interval $(- \frac{2}{3}, \infty)$ .

Increasing on $(- \frac{2}{3}, \infty)$ 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 $(- 2, 48), (- \frac{2}{3}, \frac{1040}{27})$ .

$(- 2, 48), (- \frac{2}{3}, \frac{1040}{27})$

Step 9