Calculus Examples

$\frac{1}{6} x^{4} - 2 x^{3} - 7 x^{2}$

Step 1

Write $\frac{1}{6} x^{4} - 2 x^{3} - 7 x^{2}$ as a function.

$f (x) = \frac{1}{6} x^{4} - 2 x^{3} - 7 x^{2}$

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

By the Sum Rule, the derivative of $\frac{1}{6} x^{4} - 2 x^{3} - 7 x^{2}$ with respect to $x$ is $\frac{d}{d x} [\frac{1}{6} x^{4}] + \frac{d}{d x} [- 2 x^{3}] + \frac{d}{d x} [- 7 x^{2}]$ .

$\frac{d}{d x} [\frac{1}{6} x^{4}] + \frac{d}{d x} [- 2 x^{3}] + \frac{d}{d x} [- 7 x^{2}]$

Step 2.1.2

Evaluate $\frac{d}{d x} [\frac{1}{6} x^{4}]$ .

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Step 2.1.2.1

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

$\frac{1}{6} \frac{d}{d x} [x^{4}] + \frac{d}{d x} [- 2 x^{3}] + \frac{d}{d x} [- 7 x^{2}]$

Step 2.1.2.2

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

$\frac{1}{6} (4 x^{3}) + \frac{d}{d x} [- 2 x^{3}] + \frac{d}{d x} [- 7 x^{2}]$

Step 2.1.2.3

Combine $4$ and $\frac{1}{6}$ .

$\frac{4}{6} x^{3} + \frac{d}{d x} [- 2 x^{3}] + \frac{d}{d x} [- 7 x^{2}]$

Step 2.1.2.4

Combine $\frac{4}{6}$ and $x^{3}$ .

$\frac{4 x^{3}}{6} + \frac{d}{d x} [- 2 x^{3}] + \frac{d}{d x} [- 7 x^{2}]$

Step 2.1.2.5

Cancel the common factor of $4$ and $6$ .

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Step 2.1.2.5.1

Factor $2$ out of $4 x^{3}$ .

$\frac{2 (2 x^{3})}{6} + \frac{d}{d x} [- 2 x^{3}] + \frac{d}{d x} [- 7 x^{2}]$

Step 2.1.2.5.2

Cancel the common factors.

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Step 2.1.2.5.2.1

Factor $2$ out of $6$ .

$\frac{2 (2 x^{3})}{2 (3)} + \frac{d}{d x} [- 2 x^{3}] + \frac{d}{d x} [- 7 x^{2}]$

Step 2.1.2.5.2.2

Cancel the common factor.

$\frac{2 (2 x^{3})}{2 \cdot 3} + \frac{d}{d x} [- 2 x^{3}] + \frac{d}{d x} [- 7 x^{2}]$

Step 2.1.2.5.2.3

Rewrite the expression.

$\frac{2 x^{3}}{3} + \frac{d}{d x} [- 2 x^{3}] + \frac{d}{d x} [- 7 x^{2}]$

Step 2.1.3

Evaluate $\frac{d}{d x} [- 2 x^{3}]$ .

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Step 2.1.3.1

Since $- 2$ is constant with respect to $x$ , the derivative of $- 2 x^{3}$ with respect to $x$ is $- 2 \frac{d}{d x} [x^{3}]$ .

$\frac{2 x^{3}}{3} - 2 \frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 7 x^{2}]$

Step 2.1.3.2

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

$\frac{2 x^{3}}{3} - 2 (3 x^{2}) + \frac{d}{d x} [- 7 x^{2}]$

Step 2.1.3.3

Multiply $3$ by $- 2$ .

$\frac{2 x^{3}}{3} - 6 x^{2} + \frac{d}{d x} [- 7 x^{2}]$

Step 2.1.4

Evaluate $\frac{d}{d x} [- 7 x^{2}]$ .

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Step 2.1.4.1

Since $- 7$ is constant with respect to $x$ , the derivative of $- 7 x^{2}$ with respect to $x$ is $- 7 \frac{d}{d x} [x^{2}]$ .

$\frac{2 x^{3}}{3} - 6 x^{2} - 7 \frac{d}{d x} [x^{2}]$

Step 2.1.4.2

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

$\frac{2 x^{3}}{3} - 6 x^{2} - 7 (2 x)$

Step 2.1.4.3

Multiply $2$ by $- 7$ .

$f' (x) = \frac{2 x^{3}}{3} - 6 x^{2} - 14 x$

Step 2.2

Find the second derivative.

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Step 2.2.1

By the Sum Rule, the derivative of $\frac{2 x^{3}}{3} - 6 x^{2} - 14 x$ with respect to $x$ is $\frac{d}{d x} [\frac{2 x^{3}}{3}] + \frac{d}{d x} [- 6 x^{2}] + \frac{d}{d x} [- 14 x]$ .

$\frac{d}{d x} [\frac{2 x^{3}}{3}] + \frac{d}{d x} [- 6 x^{2}] + \frac{d}{d x} [- 14 x]$

Step 2.2.2

Evaluate $\frac{d}{d x} [\frac{2 x^{3}}{3}]$ .

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Step 2.2.2.1

Since $\frac{2}{3}$ is constant with respect to $x$ , the derivative of $\frac{2 x^{3}}{3}$ with respect to $x$ is $\frac{2}{3} \frac{d}{d x} [x^{3}]$ .

$\frac{2}{3} \frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 6 x^{2}] + \frac{d}{d x} [- 14 x]$

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 $n = 3$ .

$\frac{2}{3} (3 x^{2}) + \frac{d}{d x} [- 6 x^{2}] + \frac{d}{d x} [- 14 x]$

Step 2.2.2.3

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

$\frac{3 \cdot 2}{3} x^{2} + \frac{d}{d x} [- 6 x^{2}] + \frac{d}{d x} [- 14 x]$

Step 2.2.2.4

Multiply $3$ by $2$ .

$\frac{6}{3} x^{2} + \frac{d}{d x} [- 6 x^{2}] + \frac{d}{d x} [- 14 x]$

Step 2.2.2.5

Combine $\frac{6}{3}$ and $x^{2}$ .

$\frac{6 x^{2}}{3} + \frac{d}{d x} [- 6 x^{2}] + \frac{d}{d x} [- 14 x]$

Step 2.2.2.6

Cancel the common factor of $6$ and $3$ .

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Step 2.2.2.6.1

Factor $3$ out of $6 x^{2}$ .

$\frac{3 (2 x^{2})}{3} + \frac{d}{d x} [- 6 x^{2}] + \frac{d}{d x} [- 14 x]$

Step 2.2.2.6.2

Cancel the common factors.

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Step 2.2.2.6.2.1

Factor $3$ out of $3$ .

$\frac{3 (2 x^{2})}{3 (1)} + \frac{d}{d x} [- 6 x^{2}] + \frac{d}{d x} [- 14 x]$

Step 2.2.2.6.2.2

Cancel the common factor.

$\frac{3 (2 x^{2})}{3 \cdot 1} + \frac{d}{d x} [- 6 x^{2}] + \frac{d}{d x} [- 14 x]$

Step 2.2.2.6.2.3

Rewrite the expression.

$\frac{2 x^{2}}{1} + \frac{d}{d x} [- 6 x^{2}] + \frac{d}{d x} [- 14 x]$

Step 2.2.2.6.2.4

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

$2 x^{2} + \frac{d}{d x} [- 6 x^{2}] + \frac{d}{d x} [- 14 x]$

Step 2.2.3

Evaluate $\frac{d}{d x} [- 6 x^{2}]$ .

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Step 2.2.3.1

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

$2 x^{2} - 6 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 14 x]$

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 $n = 2$ .

$2 x^{2} - 6 (2 x) + \frac{d}{d x} [- 14 x]$

Step 2.2.3.3

Multiply $2$ by $- 6$ .

$2 x^{2} - 12 x + \frac{d}{d x} [- 14 x]$

Step 2.2.4

Evaluate $\frac{d}{d x} [- 14 x]$ .

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Step 2.2.4.1

Since $- 14$ is constant with respect to $x$ , the derivative of $- 14 x$ with respect to $x$ is $- 14 \frac{d}{d x} [x]$ .

$2 x^{2} - 12 x - 14 \frac{d}{d x} [x]$

Step 2.2.4.2

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

$2 x^{2} - 12 x - 14 \cdot 1$

Step 2.2.4.3

Multiply $- 14$ by $1$ .

$f'' (x) = 2 x^{2} - 12 x - 14$

Step 2.3

The second derivative of $f (x)$ with respect to $x$ is $2 x^{2} - 12 x - 14$ .

$2 x^{2} - 12 x - 14$

Step 3

Set the second derivative equal to $0$ then solve the equation $2 x^{2} - 12 x - 14 = 0$ .

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Step 3.1

Set the second derivative equal to $0$ .

$2 x^{2} - 12 x - 14 = 0$

Step 3.2

Factor the left side of the equation.

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Step 3.2.1

Factor $2$ out of $2 x^{2} - 12 x - 14$ .

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Step 3.2.1.1

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

$2 (x^{2}) - 12 x - 14 = 0$

Step 3.2.1.2

Factor $2$ out of $- 12 x$ .

$2 (x^{2}) + 2 (- 6 x) - 14 = 0$

Step 3.2.1.3

Factor $2$ out of $- 14$ .

$2 x^{2} + 2 (- 6 x) + 2 \cdot - 7 = 0$

Step 3.2.1.4

Factor $2$ out of $2 x^{2} + 2 (- 6 x)$ .

$2 (x^{2} - 6 x) + 2 \cdot - 7 = 0$

Step 3.2.1.5

Factor $2$ out of $2 (x^{2} - 6 x) + 2 \cdot - 7$ .

$2 (x^{2} - 6 x - 7) = 0$

Step 3.2.2

Factor.

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Step 3.2.2.1

Factor $x^{2} - 6 x - 7$ using the AC method.

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Step 3.2.2.1.1

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 $- 7$ and whose sum is $- 6$ .

$- 7, 1$

Step 3.2.2.1.2

Write the factored form using these integers.

$2 ((x - 7) (x + 1)) = 0$

Step 3.2.2.2

Remove unnecessary parentheses.

$2 (x - 7) (x + 1) = 0$

Step 3.3

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

$x - 7 = 0$

$x + 1 = 0$

Step 3.4

Set $x - 7$ equal to $0$ and solve for $x$ .

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Step 3.4.1

Set $x - 7$ equal to $0$ .

$x - 7 = 0$

Step 3.4.2

Add $7$ to both sides of the equation.

$x = 7$

Step 3.5

Set $x + 1$ equal to $0$ and solve for $x$ .

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Step 3.5.1

Set $x + 1$ equal to $0$ .

$x + 1 = 0$

Step 3.5.2

Subtract $1$ from both sides of the equation.

$x = - 1$

Step 3.6

The final solution is all the values that make $2 (x - 7) (x + 1) = 0$ true.

$x = 7, - 1$

Step 4

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

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Step 4.1

Substitute $7$ in $f (x) = \frac{1}{6} x^{4} - 2 x^{3} - 7 x^{2}$ to find the value of $y$ .

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Step 4.1.1

Replace the variable $x$ with $7$ in the expression.

$f (7) = \frac{1}{6} \cdot {(7)}^{4} - 2 {(7)}^{3} - 7 {(7)}^{2}$

Step 4.1.2

Simplify the result.

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Step 4.1.2.1

Simplify each term.

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Step 4.1.2.1.1

Raise $7$ to the power of $4$ .

$f (7) = \frac{1}{6} \cdot 2401 - 2 {(7)}^{3} - 7 {(7)}^{2}$

Step 4.1.2.1.2

Combine $\frac{1}{6}$ and $2401$ .

$f (7) = \frac{2401}{6} - 2 {(7)}^{3} - 7 {(7)}^{2}$

Step 4.1.2.1.3

Raise $7$ to the power of $3$ .

$f (7) = \frac{2401}{6} - 2 \cdot 343 - 7 {(7)}^{2}$

Step 4.1.2.1.4

Multiply $- 2$ by $343$ .

$f (7) = \frac{2401}{6} - 686 - 7 {(7)}^{2}$

Step 4.1.2.1.5

Raise $7$ to the power of $2$ .

$f (7) = \frac{2401}{6} - 686 - 7 \cdot 49$

Step 4.1.2.1.6

Multiply $- 7$ by $49$ .

$f (7) = \frac{2401}{6} - 686 - 343$

Step 4.1.2.2

Find the common denominator.

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Step 4.1.2.2.1

Write $- 686$ as a fraction with denominator $1$ .

$f (7) = \frac{2401}{6} + \frac{- 686}{1} - 343$

Step 4.1.2.2.2

Multiply $\frac{- 686}{1}$ by $\frac{6}{6}$ .

$f (7) = \frac{2401}{6} + \frac{- 686}{1} \cdot \frac{6}{6} - 343$

Step 4.1.2.2.3

Multiply $\frac{- 686}{1}$ by $\frac{6}{6}$ .

$f (7) = \frac{2401}{6} + \frac{- 686 \cdot 6}{6} - 343$

Step 4.1.2.2.4

Write $- 343$ as a fraction with denominator $1$ .

$f (7) = \frac{2401}{6} + \frac{- 686 \cdot 6}{6} + \frac{- 343}{1}$

Step 4.1.2.2.5

Multiply $\frac{- 343}{1}$ by $\frac{6}{6}$ .

$f (7) = \frac{2401}{6} + \frac{- 686 \cdot 6}{6} + \frac{- 343}{1} \cdot \frac{6}{6}$

Step 4.1.2.2.6

Multiply $\frac{- 343}{1}$ by $\frac{6}{6}$ .

$f (7) = \frac{2401}{6} + \frac{- 686 \cdot 6}{6} + \frac{- 343 \cdot 6}{6}$

Step 4.1.2.3

Combine the numerators over the common denominator.

$f (7) = \frac{2401 - 686 \cdot 6 - 343 \cdot 6}{6}$

Step 4.1.2.4

Simplify each term.

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Step 4.1.2.4.1

Multiply $- 686$ by $6$ .

$f (7) = \frac{2401 - 4116 - 343 \cdot 6}{6}$

Step 4.1.2.4.2

Multiply $- 343$ by $6$ .

$f (7) = \frac{2401 - 4116 - 2058}{6}$

Step 4.1.2.5

Simplify the expression.

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Step 4.1.2.5.1

Subtract $4116$ from $2401$ .

$f (7) = \frac{- 1715 - 2058}{6}$

Step 4.1.2.5.2

Subtract $2058$ from $- 1715$ .

$f (7) = \frac{- 3773}{6}$

Step 4.1.2.5.3

Move the negative in front of the fraction.

$f (7) = - \frac{3773}{6}$

Step 4.1.2.6

The final answer is $- \frac{3773}{6}$ .

$- \frac{3773}{6}$

Step 4.2

The point found by substituting $7$ in $f (x) = \frac{1}{6} x^{4} - 2 x^{3} - 7 x^{2}$ is $(7, - \frac{3773}{6})$ . This point can be an inflection point.

$(7, - \frac{3773}{6})$

Step 4.3

Substitute $- 1$ in $f (x) = \frac{1}{6} x^{4} - 2 x^{3} - 7 x^{2}$ to find the value of $y$ .

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Step 4.3.1

Replace the variable $x$ with $- 1$ in the expression.

$f (- 1) = \frac{1}{6} \cdot {(- 1)}^{4} - 2 {(- 1)}^{3} - 7 {(- 1)}^{2}$

Step 4.3.2

Simplify the result.

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Step 4.3.2.1

Simplify each term.

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Step 4.3.2.1.1

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

$f (- 1) = \frac{1}{6} \cdot 1 - 2 {(- 1)}^{3} - 7 {(- 1)}^{2}$

Step 4.3.2.1.2

Multiply $\frac{1}{6}$ by $1$ .

$f (- 1) = \frac{1}{6} - 2 {(- 1)}^{3} - 7 {(- 1)}^{2}$

Step 4.3.2.1.3

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

$f (- 1) = \frac{1}{6} - 2 \cdot - 1 - 7 {(- 1)}^{2}$

Step 4.3.2.1.4

Multiply $- 2$ by $- 1$ .

$f (- 1) = \frac{1}{6} + 2 - 7 {(- 1)}^{2}$

Step 4.3.2.1.5

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

$f (- 1) = \frac{1}{6} + 2 - 7 \cdot 1$

Step 4.3.2.1.6

Multiply $- 7$ by $1$ .

$f (- 1) = \frac{1}{6} + 2 - 7$

Step 4.3.2.2

Find the common denominator.

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Step 4.3.2.2.1

Write $2$ as a fraction with denominator $1$ .

$f (- 1) = \frac{1}{6} + \frac{2}{1} - 7$

Step 4.3.2.2.2

Multiply $\frac{2}{1}$ by $\frac{6}{6}$ .

$f (- 1) = \frac{1}{6} + \frac{2}{1} \cdot \frac{6}{6} - 7$

Step 4.3.2.2.3

Multiply $\frac{2}{1}$ by $\frac{6}{6}$ .

$f (- 1) = \frac{1}{6} + \frac{2 \cdot 6}{6} - 7$

Step 4.3.2.2.4

Write $- 7$ as a fraction with denominator $1$ .

$f (- 1) = \frac{1}{6} + \frac{2 \cdot 6}{6} + \frac{- 7}{1}$

Step 4.3.2.2.5

Multiply $\frac{- 7}{1}$ by $\frac{6}{6}$ .

$f (- 1) = \frac{1}{6} + \frac{2 \cdot 6}{6} + \frac{- 7}{1} \cdot \frac{6}{6}$

Step 4.3.2.2.6

Multiply $\frac{- 7}{1}$ by $\frac{6}{6}$ .

$f (- 1) = \frac{1}{6} + \frac{2 \cdot 6}{6} + \frac{- 7 \cdot 6}{6}$

Step 4.3.2.3

Combine the numerators over the common denominator.

$f (- 1) = \frac{1 + 2 \cdot 6 - 7 \cdot 6}{6}$

Step 4.3.2.4

Simplify each term.

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Step 4.3.2.4.1

Multiply $2$ by $6$ .

$f (- 1) = \frac{1 + 12 - 7 \cdot 6}{6}$

Step 4.3.2.4.2

Multiply $- 7$ by $6$ .

$f (- 1) = \frac{1 + 12 - 42}{6}$

Step 4.3.2.5

Simplify the expression.

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Step 4.3.2.5.1

Add $1$ and $12$ .

$f (- 1) = \frac{13 - 42}{6}$

Step 4.3.2.5.2

Subtract $42$ from $13$ .

$f (- 1) = \frac{- 29}{6}$

Step 4.3.2.5.3

Move the negative in front of the fraction.

$f (- 1) = - \frac{29}{6}$

Step 4.3.2.6

The final answer is $- \frac{29}{6}$ .

$- \frac{29}{6}$

Step 4.4

The point found by substituting $- 1$ in $f (x) = \frac{1}{6} x^{4} - 2 x^{3} - 7 x^{2}$ is $(- 1, - \frac{29}{6})$ . This point can be an inflection point.

$(- 1, - \frac{29}{6})$

Step 4.5

Determine the points that could be inflection points.

$(7, - \frac{3773}{6}), (- 1, - \frac{29}{6})$

Step 5

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

$(- \infty, - 1) \cup (- 1, 7) \cup (7, \infty)$

Step 6

Substitute a value from the interval $(- \infty, - 1)$ into the second derivative to determine if it is increasing or decreasing.

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Step 6.1

Replace the variable $x$ with $- 1.1$ in the expression.

$f'' (- 1.1) = 2 {(- 1.1)}^{2} - 12 \cdot - 1.1 - 14$

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 $- 1.1$ to the power of $2$ .

$f'' (- 1.1) = 2 \cdot 1.21 - 12 \cdot - 1.1 - 14$

Step 6.2.1.2

Multiply $2$ by $1.21$ .

$f'' (- 1.1) = 2.42 - 12 \cdot - 1.1 - 14$

Step 6.2.1.3

Multiply $- 12$ by $- 1.1$ .

$f'' (- 1.1) = 2.42 + 13.2 - 14$

Step 6.2.2

Simplify by adding and subtracting.

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Step 6.2.2.1

Add $2.42$ and $13.2$ .

$f'' (- 1.1) = 15.62 - 14$

Step 6.2.2.2

Subtract $14$ from $15.62$ .

$f'' (- 1.1) = 1.62$

Step 6.2.3

The final answer is $1.62$ .

$1.62$

Step 6.3

At $- 1.1$ , the second derivative is $1.62$ . Since this is positive, the second derivative is increasing on the interval $(- \infty, - 1)$ .

Increasing on $(- \infty, - 1)$ since $f'' (x) > 0$

Step 7

Substitute a value from the interval $(- 1, 7)$ into the second derivative to determine if it is increasing or decreasing.

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Step 7.1

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

$f'' (3) = 2 {(3)}^{2} - 12 \cdot 3 - 14$

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 $3$ to the power of $2$ .

$f'' (3) = 2 \cdot 9 - 12 \cdot 3 - 14$

Step 7.2.1.2

Multiply $2$ by $9$ .

$f'' (3) = 18 - 12 \cdot 3 - 14$

Step 7.2.1.3

Multiply $- 12$ by $3$ .

$f'' (3) = 18 - 36 - 14$

Step 7.2.2

Simplify by subtracting numbers.

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Step 7.2.2.1

Subtract $36$ from $18$ .

$f'' (3) = - 18 - 14$

Step 7.2.2.2

Subtract $14$ from $- 18$ .

$f'' (3) = - 32$

Step 7.2.3

The final answer is $- 32$ .

$- 32$

Step 7.3

At $3$ , the second derivative is $- 32$ . Since this is negative, the second derivative is decreasing on the interval $(- 1, 7)$

Decreasing on $(- 1, 7)$ since $f'' (x) < 0$

Step 8

Substitute a value from the interval $(7, \infty)$ into the second derivative to determine if it is increasing or decreasing.

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Step 8.1

Replace the variable $x$ with $7.1$ in the expression.

$f'' (7.1) = 2 {(7.1)}^{2} - 12 \cdot 7.1 - 14$

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 $7.1$ to the power of $2$ .

$f'' (7.1) = 2 \cdot 50.41 - 12 \cdot 7.1 - 14$

Step 8.2.1.2

Multiply $2$ by $50.41$ .

$f'' (7.1) = 100.82 - 12 \cdot 7.1 - 14$

Step 8.2.1.3

Multiply $- 12$ by $7.1$ .

$f'' (7.1) = 100.82 - 85.2 - 14$

Step 8.2.2

Simplify by subtracting numbers.

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Step 8.2.2.1

Subtract $85.2$ from $100.82$ .

$f'' (7.1) = 15.62 - 14$

Step 8.2.2.2

Subtract $14$ from $15.62$ .

$f'' (7.1) = 1.62$

Step 8.2.3

The final answer is $1.62$ .

$1.62$

Step 8.3

At $7.1$ , the second derivative is $1.62$ . Since this is positive, the second derivative is increasing on the interval $(7, \infty)$ .

Increasing on $(7, \infty)$ 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 $(- 1, - \frac{29}{6}), (7, - \frac{3773}{6})$ .

$(- 1, - \frac{29}{6}), (7, - \frac{3773}{6})$

Step 10