Calculus Examples

$f (x) = \frac{1}{12} x^{4} + 3 x^{3} + 40 x^{2}$

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 $\frac{1}{12} x^{4} + 3 x^{3} + 40 x^{2}$ with respect to $x$ is $\frac{d}{d x} [\frac{1}{12} x^{4}] + \frac{d}{d x} [3 x^{3}] + \frac{d}{d x} [40 x^{2}]$ .

$\frac{d}{d x} [\frac{1}{12} x^{4}] + \frac{d}{d x} [3 x^{3}] + \frac{d}{d x} [40 x^{2}]$

Step 1.1.2

Evaluate $\frac{d}{d x} [\frac{1}{12} x^{4}]$ .

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Step 1.1.2.1

Since $\frac{1}{12}$ is constant with respect to $x$ , the derivative of $\frac{1}{12} x^{4}$ with respect to $x$ is $\frac{1}{12} \frac{d}{d x} [x^{4}]$ .

$\frac{1}{12} \frac{d}{d x} [x^{4}] + \frac{d}{d x} [3 x^{3}] + \frac{d}{d x} [40 x^{2}]$

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 $n = 4$ .

$\frac{1}{12} (4 x^{3}) + \frac{d}{d x} [3 x^{3}] + \frac{d}{d x} [40 x^{2}]$

Step 1.1.2.3

Combine $4$ and $\frac{1}{12}$ .

$\frac{4}{12} x^{3} + \frac{d}{d x} [3 x^{3}] + \frac{d}{d x} [40 x^{2}]$

Step 1.1.2.4

Combine $\frac{4}{12}$ and $x^{3}$ .

$\frac{4 x^{3}}{12} + \frac{d}{d x} [3 x^{3}] + \frac{d}{d x} [40 x^{2}]$

Step 1.1.2.5

Cancel the common factor of $4$ and $12$ .

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Step 1.1.2.5.1

Factor $4$ out of $4 x^{3}$ .

$\frac{4 (x^{3})}{12} + \frac{d}{d x} [3 x^{3}] + \frac{d}{d x} [40 x^{2}]$

Step 1.1.2.5.2

Cancel the common factors.

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Step 1.1.2.5.2.1

Factor $4$ out of $12$ .

$\frac{4 x^{3}}{4 \cdot 3} + \frac{d}{d x} [3 x^{3}] + \frac{d}{d x} [40 x^{2}]$

Step 1.1.2.5.2.2

Cancel the common factor.

$\frac{4 x^{3}}{4 \cdot 3} + \frac{d}{d x} [3 x^{3}] + \frac{d}{d x} [40 x^{2}]$

Step 1.1.2.5.2.3

Rewrite the expression.

$\frac{x^{3}}{3} + \frac{d}{d x} [3 x^{3}] + \frac{d}{d x} [40 x^{2}]$

Step 1.1.3

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

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Step 1.1.3.1

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

$\frac{x^{3}}{3} + 3 \frac{d}{d x} [x^{3}] + \frac{d}{d x} [40 x^{2}]$

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 $n = 3$ .

$\frac{x^{3}}{3} + 3 (3 x^{2}) + \frac{d}{d x} [40 x^{2}]$

Step 1.1.3.3

Multiply $3$ by $3$ .

$\frac{x^{3}}{3} + 9 x^{2} + \frac{d}{d x} [40 x^{2}]$

Step 1.1.4

Evaluate $\frac{d}{d x} [40 x^{2}]$ .

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Step 1.1.4.1

Since $40$ is constant with respect to $x$ , the derivative of $40 x^{2}$ with respect to $x$ is $40 \frac{d}{d x} [x^{2}]$ .

$\frac{x^{3}}{3} + 9 x^{2} + 40 \frac{d}{d x} [x^{2}]$

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 $n = 2$ .

$\frac{x^{3}}{3} + 9 x^{2} + 40 (2 x)$

Step 1.1.4.3

Multiply $2$ by $40$ .

$f' (x) = \frac{x^{3}}{3} + 9 x^{2} + 80 x$

Step 1.2

Find the second derivative.

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Step 1.2.1

By the Sum Rule, the derivative of $\frac{x^{3}}{3} + 9 x^{2} + 80 x$ with respect to $x$ is $\frac{d}{d x} [\frac{x^{3}}{3}] + \frac{d}{d x} [9 x^{2}] + \frac{d}{d x} [80 x]$ .

$\frac{d}{d x} [\frac{x^{3}}{3}] + \frac{d}{d x} [9 x^{2}] + \frac{d}{d x} [80 x]$

Step 1.2.2

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

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Step 1.2.2.1

Since $\frac{1}{3}$ is constant with respect to $x$ , the derivative of $\frac{x^{3}}{3}$ with respect to $x$ is $\frac{1}{3} \frac{d}{d x} [x^{3}]$ .

$\frac{1}{3} \frac{d}{d x} [x^{3}] + \frac{d}{d x} [9 x^{2}] + \frac{d}{d x} [80 x]$

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 $n = 3$ .

$\frac{1}{3} (3 x^{2}) + \frac{d}{d x} [9 x^{2}] + \frac{d}{d x} [80 x]$

Step 1.2.2.3

Combine $3$ and $\frac{1}{3}$ .

$\frac{3}{3} x^{2} + \frac{d}{d x} [9 x^{2}] + \frac{d}{d x} [80 x]$

Step 1.2.2.4

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

$\frac{3 x^{2}}{3} + \frac{d}{d x} [9 x^{2}] + \frac{d}{d x} [80 x]$

Step 1.2.2.5

Cancel the common factor of $3$ .

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Step 1.2.2.5.1

Cancel the common factor.

$\frac{3 x^{2}}{3} + \frac{d}{d x} [9 x^{2}] + \frac{d}{d x} [80 x]$

Step 1.2.2.5.2

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

$x^{2} + \frac{d}{d x} [9 x^{2}] + \frac{d}{d x} [80 x]$

Step 1.2.3

Evaluate $\frac{d}{d x} [9 x^{2}]$ .

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Step 1.2.3.1

Since $9$ is constant with respect to $x$ , the derivative of $9 x^{2}$ with respect to $x$ is $9 \frac{d}{d x} [x^{2}]$ .

$x^{2} + 9 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [80 x]$

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 $n = 2$ .

$x^{2} + 9 (2 x) + \frac{d}{d x} [80 x]$

Step 1.2.3.3

Multiply $2$ by $9$ .

$x^{2} + 18 x + \frac{d}{d x} [80 x]$

Step 1.2.4

Evaluate $\frac{d}{d x} [80 x]$ .

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Step 1.2.4.1

Since $80$ is constant with respect to $x$ , the derivative of $80 x$ with respect to $x$ is $80 \frac{d}{d x} [x]$ .

$x^{2} + 18 x + 80 \frac{d}{d x} [x]$

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 $n = 1$ .

$x^{2} + 18 x + 80 \cdot 1$

Step 1.2.4.3

Multiply $80$ by $1$ .

$f'' (x) = x^{2} + 18 x + 80$

Step 1.3

The second derivative of $f (x)$ with respect to $x$ is $x^{2} + 18 x + 80$ .

$x^{2} + 18 x + 80$

Step 2

Set the second derivative equal to $0$ then solve the equation $x^{2} + 18 x + 80 = 0$ .

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Step 2.1

Set the second derivative equal to $0$ .

$x^{2} + 18 x + 80 = 0$

Step 2.2

Factor $x^{2} + 18 x + 80$ using the AC method.

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Step 2.2.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $80$ and whose sum is $18$ .

$8, 10$

Step 2.2.2

Write the factored form using these integers.

$(x + 8) (x + 10) = 0$

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 $0$ .

$x + 8 = 0$

$x + 10 = 0$

Step 2.4

Set $x + 8$ equal to $0$ and solve for $x$ .

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Step 2.4.1

Set $x + 8$ equal to $0$ .

$x + 8 = 0$

Step 2.4.2

Subtract $8$ from both sides of the equation.

$x = - 8$

Step 2.5

Set $x + 10$ equal to $0$ and solve for $x$ .

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Step 2.5.1

Set $x + 10$ equal to $0$ .

$x + 10 = 0$

Step 2.5.2

Subtract $10$ from both sides of the equation.

$x = - 10$

Step 2.6

The final solution is all the values that make $(x + 8) (x + 10) = 0$ true.

$x = - 8, - 10$

Step 3

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

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Step 3.1

Substitute $- 8$ in $f (x) = \frac{1}{12} x^{4} + 3 x^{3} + 40 x^{2}$ to find the value of $y$ .

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Step 3.1.1

Replace the variable $x$ with $- 8$ in the expression.

$f (- 8) = \frac{1}{12} \cdot {(- 8)}^{4} + 3 {(- 8)}^{3} + 40 {(- 8)}^{2}$

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

Raise $- 8$ to the power of $4$ .

$f (- 8) = \frac{1}{12} \cdot 4096 + 3 {(- 8)}^{3} + 40 {(- 8)}^{2}$

Step 3.1.2.1.2

Cancel the common factor of $4$ .

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Step 3.1.2.1.2.1

Factor $4$ out of $12$ .

$f (- 8) = \frac{1}{4 (3)} \cdot 4096 + 3 {(- 8)}^{3} + 40 {(- 8)}^{2}$

Step 3.1.2.1.2.2

Factor $4$ out of $4096$ .

$f (- 8) = \frac{1}{4 \cdot 3} \cdot (4 \cdot 1024) + 3 {(- 8)}^{3} + 40 {(- 8)}^{2}$

Step 3.1.2.1.2.3

Cancel the common factor.

$f (- 8) = \frac{1}{4 \cdot 3} \cdot (4 \cdot 1024) + 3 {(- 8)}^{3} + 40 {(- 8)}^{2}$

Step 3.1.2.1.2.4

Rewrite the expression.

$f (- 8) = \frac{1}{3} \cdot 1024 + 3 {(- 8)}^{3} + 40 {(- 8)}^{2}$

Step 3.1.2.1.3

Combine $\frac{1}{3}$ and $1024$ .

$f (- 8) = \frac{1024}{3} + 3 {(- 8)}^{3} + 40 {(- 8)}^{2}$

Step 3.1.2.1.4

Raise $- 8$ to the power of $3$ .

$f (- 8) = \frac{1024}{3} + 3 \cdot - 512 + 40 {(- 8)}^{2}$

Step 3.1.2.1.5

Multiply $3$ by $- 512$ .

$f (- 8) = \frac{1024}{3} - 1536 + 40 {(- 8)}^{2}$

Step 3.1.2.1.6

Raise $- 8$ to the power of $2$ .

$f (- 8) = \frac{1024}{3} - 1536 + 40 \cdot 64$

Step 3.1.2.1.7

Multiply $40$ by $64$ .

$f (- 8) = \frac{1024}{3} - 1536 + 2560$

Step 3.1.2.2

Find the common denominator.

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Step 3.1.2.2.1

Write $- 1536$ as a fraction with denominator $1$ .

$f (- 8) = \frac{1024}{3} + \frac{- 1536}{1} + 2560$

Step 3.1.2.2.2

Multiply $\frac{- 1536}{1}$ by $\frac{3}{3}$ .

$f (- 8) = \frac{1024}{3} + \frac{- 1536}{1} \cdot \frac{3}{3} + 2560$

Step 3.1.2.2.3

Multiply $\frac{- 1536}{1}$ by $\frac{3}{3}$ .

$f (- 8) = \frac{1024}{3} + \frac{- 1536 \cdot 3}{3} + 2560$

Step 3.1.2.2.4

Write $2560$ as a fraction with denominator $1$ .

$f (- 8) = \frac{1024}{3} + \frac{- 1536 \cdot 3}{3} + \frac{2560}{1}$

Step 3.1.2.2.5

Multiply $\frac{2560}{1}$ by $\frac{3}{3}$ .

$f (- 8) = \frac{1024}{3} + \frac{- 1536 \cdot 3}{3} + \frac{2560}{1} \cdot \frac{3}{3}$

Step 3.1.2.2.6

Multiply $\frac{2560}{1}$ by $\frac{3}{3}$ .

$f (- 8) = \frac{1024}{3} + \frac{- 1536 \cdot 3}{3} + \frac{2560 \cdot 3}{3}$

Step 3.1.2.3

Combine the numerators over the common denominator.

$f (- 8) = \frac{1024 - 1536 \cdot 3 + 2560 \cdot 3}{3}$

Step 3.1.2.4

Simplify each term.

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Step 3.1.2.4.1

Multiply $- 1536$ by $3$ .

$f (- 8) = \frac{1024 - 4608 + 2560 \cdot 3}{3}$

Step 3.1.2.4.2

Multiply $2560$ by $3$ .

$f (- 8) = \frac{1024 - 4608 + 7680}{3}$

Step 3.1.2.5

Simplify by adding and subtracting.

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Step 3.1.2.5.1

Subtract $4608$ from $1024$ .

$f (- 8) = \frac{- 3584 + 7680}{3}$

Step 3.1.2.5.2

Add $- 3584$ and $7680$ .

$f (- 8) = \frac{4096}{3}$

Step 3.1.2.6

The final answer is $\frac{4096}{3}$ .

$\frac{4096}{3}$

Step 3.2

The point found by substituting $- 8$ in $f (x) = \frac{1}{12} x^{4} + 3 x^{3} + 40 x^{2}$ is $(- 8, \frac{4096}{3})$ . This point can be an inflection point.

$(- 8, \frac{4096}{3})$

Step 3.3

Substitute $- 10$ in $f (x) = \frac{1}{12} x^{4} + 3 x^{3} + 40 x^{2}$ to find the value of $y$ .

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Step 3.3.1

Replace the variable $x$ with $- 10$ in the expression.

$f (- 10) = \frac{1}{12} \cdot {(- 10)}^{4} + 3 {(- 10)}^{3} + 40 {(- 10)}^{2}$

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 $- 10$ to the power of $4$ .

$f (- 10) = \frac{1}{12} \cdot 10000 + 3 {(- 10)}^{3} + 40 {(- 10)}^{2}$

Step 3.3.2.1.2

Cancel the common factor of $4$ .

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Step 3.3.2.1.2.1

Factor $4$ out of $12$ .

$f (- 10) = \frac{1}{4 (3)} \cdot 10000 + 3 {(- 10)}^{3} + 40 {(- 10)}^{2}$

Step 3.3.2.1.2.2

Factor $4$ out of $10000$ .

$f (- 10) = \frac{1}{4 \cdot 3} \cdot (4 \cdot 2500) + 3 {(- 10)}^{3} + 40 {(- 10)}^{2}$

Step 3.3.2.1.2.3

Cancel the common factor.

$f (- 10) = \frac{1}{4 \cdot 3} \cdot (4 \cdot 2500) + 3 {(- 10)}^{3} + 40 {(- 10)}^{2}$

Step 3.3.2.1.2.4

Rewrite the expression.

$f (- 10) = \frac{1}{3} \cdot 2500 + 3 {(- 10)}^{3} + 40 {(- 10)}^{2}$

Step 3.3.2.1.3

Combine $\frac{1}{3}$ and $2500$ .

$f (- 10) = \frac{2500}{3} + 3 {(- 10)}^{3} + 40 {(- 10)}^{2}$

Step 3.3.2.1.4

Raise $- 10$ to the power of $3$ .

$f (- 10) = \frac{2500}{3} + 3 \cdot - 1000 + 40 {(- 10)}^{2}$

Step 3.3.2.1.5

Multiply $3$ by $- 1000$ .

$f (- 10) = \frac{2500}{3} - 3000 + 40 {(- 10)}^{2}$

Step 3.3.2.1.6

Raise $- 10$ to the power of $2$ .

$f (- 10) = \frac{2500}{3} - 3000 + 40 \cdot 100$

Step 3.3.2.1.7

Multiply $40$ by $100$ .

$f (- 10) = \frac{2500}{3} - 3000 + 4000$

Step 3.3.2.2

Find the common denominator.

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Step 3.3.2.2.1

Write $- 3000$ as a fraction with denominator $1$ .

$f (- 10) = \frac{2500}{3} + \frac{- 3000}{1} + 4000$

Step 3.3.2.2.2

Multiply $\frac{- 3000}{1}$ by $\frac{3}{3}$ .

$f (- 10) = \frac{2500}{3} + \frac{- 3000}{1} \cdot \frac{3}{3} + 4000$

Step 3.3.2.2.3

Multiply $\frac{- 3000}{1}$ by $\frac{3}{3}$ .

$f (- 10) = \frac{2500}{3} + \frac{- 3000 \cdot 3}{3} + 4000$

Step 3.3.2.2.4

Write $4000$ as a fraction with denominator $1$ .

$f (- 10) = \frac{2500}{3} + \frac{- 3000 \cdot 3}{3} + \frac{4000}{1}$

Step 3.3.2.2.5

Multiply $\frac{4000}{1}$ by $\frac{3}{3}$ .

$f (- 10) = \frac{2500}{3} + \frac{- 3000 \cdot 3}{3} + \frac{4000}{1} \cdot \frac{3}{3}$

Step 3.3.2.2.6

Multiply $\frac{4000}{1}$ by $\frac{3}{3}$ .

$f (- 10) = \frac{2500}{3} + \frac{- 3000 \cdot 3}{3} + \frac{4000 \cdot 3}{3}$

Step 3.3.2.3

Combine the numerators over the common denominator.

$f (- 10) = \frac{2500 - 3000 \cdot 3 + 4000 \cdot 3}{3}$

Step 3.3.2.4

Simplify each term.

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Step 3.3.2.4.1

Multiply $- 3000$ by $3$ .

$f (- 10) = \frac{2500 - 9000 + 4000 \cdot 3}{3}$

Step 3.3.2.4.2

Multiply $4000$ by $3$ .

$f (- 10) = \frac{2500 - 9000 + 12000}{3}$

Step 3.3.2.5

Simplify by adding and subtracting.

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Step 3.3.2.5.1

Subtract $9000$ from $2500$ .

$f (- 10) = \frac{- 6500 + 12000}{3}$

Step 3.3.2.5.2

Add $- 6500$ and $12000$ .

$f (- 10) = \frac{5500}{3}$

Step 3.3.2.6

The final answer is $\frac{5500}{3}$ .

$\frac{5500}{3}$

Step 3.4

The point found by substituting $- 10$ in $f (x) = \frac{1}{12} x^{4} + 3 x^{3} + 40 x^{2}$ is $(- 10, \frac{5500}{3})$ . This point can be an inflection point.

$(- 10, \frac{5500}{3})$

Step 3.5

Determine the points that could be inflection points.

$(- 8, \frac{4096}{3}), (- 10, \frac{5500}{3})$

Step 4

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

$(- \infty, - 10) \cup (- 10, - 8) \cup (- 8, \infty)$

Step 5

Substitute a value from the interval $(- \infty, - 10)$ into the second derivative to determine if it is increasing or decreasing.

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Step 5.1

Replace the variable $x$ with $- 10.1$ in the expression.

$f'' (- 10.1) = {(- 10.1)}^{2} + 18 (- 10.1) + 80$

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 $- 10.1$ to the power of $2$ .

$f'' (- 10.1) = 102.01 + 18 (- 10.1) + 80$

Step 5.2.1.2

Multiply $18$ by $- 10.1$ .

$f'' (- 10.1) = 102.01 - 181.8 + 80$

Step 5.2.2

Simplify by adding and subtracting.

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Step 5.2.2.1

Subtract $181.8$ from $102.01$ .

$f'' (- 10.1) = - 79.79 + 80$

Step 5.2.2.2

Add $- 79.79$ and $80$ .

$f'' (- 10.1) = 0.21$

Step 5.2.3

The final answer is $0.21$ .

$0.21$

Step 5.3

At $- 10.1$ , the second derivative is $0.21$ . Since this is positive, the second derivative is increasing on the interval $(- \infty, - 10)$ .

Increasing on $(- \infty, - 10)$ since $f'' (x) > 0$

Step 6

Substitute a value from the interval $(- 10, - 8)$ into the second derivative to determine if it is increasing or decreasing.

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Step 6.1

Replace the variable $x$ with $- 9$ in the expression.

$f'' (- 9) = {(- 9)}^{2} + 18 (- 9) + 80$

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 $- 9$ to the power of $2$ .

$f'' (- 9) = 81 + 18 (- 9) + 80$

Step 6.2.1.2

Multiply $18$ by $- 9$ .

$f'' (- 9) = 81 - 162 + 80$

Step 6.2.2

Simplify by adding and subtracting.

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Step 6.2.2.1

Subtract $162$ from $81$ .

$f'' (- 9) = - 81 + 80$

Step 6.2.2.2

Add $- 81$ and $80$ .

$f'' (- 9) = - 1$

Step 6.2.3

The final answer is $- 1$ .

$- 1$

Step 6.3

At $- 9$ , the second derivative is $- 1$ . Since this is negative, the second derivative is decreasing on the interval $(- 10, - 8)$

Decreasing on $(- 10, - 8)$ since $f'' (x) < 0$

Step 7

Substitute a value from the interval $(- 8, \infty)$ into the second derivative to determine if it is increasing or decreasing.

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Step 7.1

Replace the variable $x$ with $- 7.9$ in the expression.

$f'' (- 7.9) = {(- 7.9)}^{2} + 18 (- 7.9) + 80$

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 $- 7.9$ to the power of $2$ .

$f'' (- 7.9) = 62.41 + 18 (- 7.9) + 80$

Step 7.2.1.2

Multiply $18$ by $- 7.9$ .

$f'' (- 7.9) = 62.41 - 142.2 + 80$

Step 7.2.2

Simplify by adding and subtracting.

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Step 7.2.2.1

Subtract $142.2$ from $62.41$ .

$f'' (- 7.9) = - 79.79 + 80$

Step 7.2.2.2

Add $- 79.79$ and $80$ .

$f'' (- 7.9) = 0.21$

Step 7.2.3

The final answer is $0.21$ .

$0.21$

Step 7.3

At $- 7.9$ , the second derivative is $0.21$ . Since this is positive, the second derivative is increasing on the interval $(- 8, \infty)$ .

Increasing on $(- 8, \infty)$ 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 $(- 10, \frac{5500}{3}), (- 8, \frac{4096}{3})$ .

$(- 10, \frac{5500}{3}), (- 8, \frac{4096}{3})$

Step 9