Calculus Examples

$f (x) = - \frac{1}{6} x^{4} + x^{3} + 18 x^{2}$

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 $- \frac{1}{6} x^{4} + x^{3} + 18 x^{2}$ with respect to $x$ is $\frac{d}{d x} [- \frac{1}{6} x^{4}] + \frac{d}{d x} [x^{3}] + \frac{d}{d x} [18 x^{2}]$ .

$\frac{d}{d x} [- \frac{1}{6} x^{4}] + \frac{d}{d x} [x^{3}] + \frac{d}{d x} [18 x^{2}]$

Step 1.1.2

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

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Step 1.1.2.1

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

$- \frac{1}{6} \frac{d}{d x} [x^{4}] + \frac{d}{d x} [x^{3}] + \frac{d}{d x} [18 x^{2}]$

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 $n = 4$ .

$- \frac{1}{6} (4 x^{3}) + \frac{d}{d x} [x^{3}] + \frac{d}{d x} [18 x^{2}]$

Step 1.1.2.3

Multiply $4$ by $- 1$ .

$- 4 (\frac{1}{6}) x^{3} + \frac{d}{d x} [x^{3}] + \frac{d}{d x} [18 x^{2}]$

Step 1.1.2.4

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

$\frac{- 4}{6} x^{3} + \frac{d}{d x} [x^{3}] + \frac{d}{d x} [18 x^{2}]$

Step 1.1.2.5

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

$\frac{- 4 x^{3}}{6} + \frac{d}{d x} [x^{3}] + \frac{d}{d x} [18 x^{2}]$

Step 1.1.2.6

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

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Step 1.1.2.6.1

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

$\frac{2 (- 2 x^{3})}{6} + \frac{d}{d x} [x^{3}] + \frac{d}{d x} [18 x^{2}]$

Step 1.1.2.6.2

Cancel the common factors.

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Step 1.1.2.6.2.1

Factor $2$ out of $6$ .

$\frac{2 (- 2 x^{3})}{2 (3)} + \frac{d}{d x} [x^{3}] + \frac{d}{d x} [18 x^{2}]$

Step 1.1.2.6.2.2

Cancel the common factor.

$\frac{2 (- 2 x^{3})}{2 \cdot 3} + \frac{d}{d x} [x^{3}] + \frac{d}{d x} [18 x^{2}]$

Step 1.1.2.6.2.3

Rewrite the expression.

$\frac{- 2 x^{3}}{3} + \frac{d}{d x} [x^{3}] + \frac{d}{d x} [18 x^{2}]$

Step 1.1.2.7

Move the negative in front of the fraction.

$- \frac{2 x^{3}}{3} + \frac{d}{d x} [x^{3}] + \frac{d}{d x} [18 x^{2}]$

Step 1.1.3

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

$- \frac{2 x^{3}}{3} + 3 x^{2} + \frac{d}{d x} [18 x^{2}]$

Step 1.1.4

Evaluate $\frac{d}{d x} [18 x^{2}]$ .

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Step 1.1.4.1

Since $18$ is constant with respect to $x$ , the derivative of $18 x^{2}$ with respect to $x$ is $18 \frac{d}{d x} [x^{2}]$ .

$- \frac{2 x^{3}}{3} + 3 x^{2} + 18 \frac{d}{d x} [x^{2}]$

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 $n = 2$ .

$- \frac{2 x^{3}}{3} + 3 x^{2} + 18 (2 x)$

Step 1.1.4.3

Multiply $2$ by $18$ .

$f' (x) = - \frac{2 x^{3}}{3} + 3 x^{2} + 36 x$

Step 1.2

Find the second derivative.

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Step 1.2.1

By the Sum Rule, the derivative of $- \frac{2 x^{3}}{3} + 3 x^{2} + 36 x$ with respect to $x$ is $\frac{d}{d x} [- \frac{2 x^{3}}{3}] + \frac{d}{d x} [3 x^{2}] + \frac{d}{d x} [36 x]$ .

$\frac{d}{d x} [- \frac{2 x^{3}}{3}] + \frac{d}{d x} [3 x^{2}] + \frac{d}{d x} [36 x]$

Step 1.2.2

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

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Step 1.2.2.1

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

$- \frac{2}{3} \frac{d}{d x} [x^{3}] + \frac{d}{d x} [3 x^{2}] + \frac{d}{d x} [36 x]$

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 $n = 3$ .

$- \frac{2}{3} (3 x^{2}) + \frac{d}{d x} [3 x^{2}] + \frac{d}{d x} [36 x]$

Step 1.2.2.3

Multiply $3$ by $- 1$ .

$- 3 (\frac{2}{3}) x^{2} + \frac{d}{d x} [3 x^{2}] + \frac{d}{d x} [36 x]$

Step 1.2.2.4

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

$\frac{- 3 \cdot 2}{3} x^{2} + \frac{d}{d x} [3 x^{2}] + \frac{d}{d x} [36 x]$

Step 1.2.2.5

Multiply $- 3$ by $2$ .

$\frac{- 6}{3} x^{2} + \frac{d}{d x} [3 x^{2}] + \frac{d}{d x} [36 x]$

Step 1.2.2.6

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

$\frac{- 6 x^{2}}{3} + \frac{d}{d x} [3 x^{2}] + \frac{d}{d x} [36 x]$

Step 1.2.2.7

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

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Step 1.2.2.7.1

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

$\frac{3 (- 2 x^{2})}{3} + \frac{d}{d x} [3 x^{2}] + \frac{d}{d x} [36 x]$

Step 1.2.2.7.2

Cancel the common factors.

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Step 1.2.2.7.2.1

Factor $3$ out of $3$ .

$\frac{3 (- 2 x^{2})}{3 (1)} + \frac{d}{d x} [3 x^{2}] + \frac{d}{d x} [36 x]$

Step 1.2.2.7.2.2

Cancel the common factor.

$\frac{3 (- 2 x^{2})}{3 \cdot 1} + \frac{d}{d x} [3 x^{2}] + \frac{d}{d x} [36 x]$

Step 1.2.2.7.2.3

Rewrite the expression.

$\frac{- 2 x^{2}}{1} + \frac{d}{d x} [3 x^{2}] + \frac{d}{d x} [36 x]$

Step 1.2.2.7.2.4

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

$- 2 x^{2} + \frac{d}{d x} [3 x^{2}] + \frac{d}{d x} [36 x]$

Step 1.2.3

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

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Step 1.2.3.1

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

$- 2 x^{2} + 3 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [36 x]$

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 $n = 2$ .

$- 2 x^{2} + 3 (2 x) + \frac{d}{d x} [36 x]$

Step 1.2.3.3

Multiply $2$ by $3$ .

$- 2 x^{2} + 6 x + \frac{d}{d x} [36 x]$

Step 1.2.4

Evaluate $\frac{d}{d x} [36 x]$ .

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Step 1.2.4.1

Since $36$ is constant with respect to $x$ , the derivative of $36 x$ with respect to $x$ is $36 \frac{d}{d x} [x]$ .

$- 2 x^{2} + 6 x + 36 \frac{d}{d x} [x]$

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 $n = 1$ .

$- 2 x^{2} + 6 x + 36 \cdot 1$

Step 1.2.4.3

Multiply $36$ by $1$ .

$f'' (x) = - 2 x^{2} + 6 x + 36$

Step 1.3

The second derivative of $f (x)$ with respect to $x$ is $- 2 x^{2} + 6 x + 36$ .

$- 2 x^{2} + 6 x + 36$

Step 2

Set the second derivative equal to $0$ then solve the equation $- 2 x^{2} + 6 x + 36 = 0$ .

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Step 2.1

Set the second derivative equal to $0$ .

$- 2 x^{2} + 6 x + 36 = 0$

Step 2.2

Factor the left side of the equation.

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Step 2.2.1

Factor $- 2$ out of $- 2 x^{2} + 6 x + 36$ .

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Step 2.2.1.1

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

$- 2 x^{2} + 6 x + 36 = 0$

Step 2.2.1.2

Factor $- 2$ out of $6 x$ .

$- 2 x^{2} - 2 (- 3 x) + 36 = 0$

Step 2.2.1.3

Factor $- 2$ out of $36$ .

$- 2 x^{2} - 2 (- 3 x) - 2 \cdot - 18 = 0$

Step 2.2.1.4

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

$- 2 (x^{2} - 3 x) - 2 \cdot - 18 = 0$

Step 2.2.1.5

Factor $- 2$ out of $- 2 (x^{2} - 3 x) - 2 (- 18)$ .

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

Step 2.2.2

Factor.

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Step 2.2.2.1

Factor $x^{2} - 3 x - 18$ using the AC method.

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Step 2.2.2.1.1

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 $- 18$ and whose sum is $- 3$ .

$- 6, 3$

Step 2.2.2.1.2

Write the factored form using these integers.

$- 2 ((x - 6) (x + 3)) = 0$

Step 2.2.2.2

Remove unnecessary parentheses.

$- 2 (x - 6) (x + 3) = 0$

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 $0$ .

$x - 6 = 0$

$x + 3 = 0$

Step 2.4

Set $x - 6$ equal to $0$ and solve for $x$ .

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Step 2.4.1

Set $x - 6$ equal to $0$ .

$x - 6 = 0$

Step 2.4.2

Add $6$ to both sides of the equation.

$x = 6$

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

Subtract $3$ from both sides of the equation.

$x = - 3$

Step 2.6

The final solution is all the values that make $- 2 (x - 6) (x + 3) = 0$ true.

$x = 6, - 3$

Step 3

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

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Step 3.1

Substitute $6$ in $f (x) = - \frac{1}{6} x^{4} + x^{3} + 18 x^{2}$ to find the value of $y$ .

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Step 3.1.1

Replace the variable $x$ with $6$ in the expression.

$f (6) = - \frac{1}{6} \cdot {(6)}^{4} + {(6)}^{3} + 18 {(6)}^{2}$

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

Cancel the common factor of $6$ .

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Step 3.1.2.1.1.1

Move the leading negative in $- \frac{1}{6}$ into the numerator.

$f (6) = \frac{- 1}{6} \cdot {(6)}^{4} + {(6)}^{3} + 18 {(6)}^{2}$

Step 3.1.2.1.1.2

Factor $6$ out of ${(6)}^{4}$ .

$f (6) = \frac{- 1}{6} \cdot (6 \cdot 6^{3}) + {(6)}^{3} + 18 {(6)}^{2}$

Step 3.1.2.1.1.3

Cancel the common factor.

$f (6) = \frac{- 1}{6} \cdot (6 \cdot 6^{3}) + {(6)}^{3} + 18 {(6)}^{2}$

Step 3.1.2.1.1.4

Rewrite the expression.

$f (6) = - 1 \cdot 6^{3} + {(6)}^{3} + 18 {(6)}^{2}$

Step 3.1.2.1.2

Raise $6$ to the power of $3$ .

$f (6) = - 1 \cdot 216 + {(6)}^{3} + 18 {(6)}^{2}$

Step 3.1.2.1.3

Multiply $- 1$ by $216$ .

$f (6) = - 216 + {(6)}^{3} + 18 {(6)}^{2}$

Step 3.1.2.1.4

Raise $6$ to the power of $3$ .

$f (6) = - 216 + 216 + 18 {(6)}^{2}$

Step 3.1.2.1.5

Raise $6$ to the power of $2$ .

$f (6) = - 216 + 216 + 18 \cdot 36$

Step 3.1.2.1.6

Multiply $18$ by $36$ .

$f (6) = - 216 + 216 + 648$

Step 3.1.2.2

Simplify by adding numbers.

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Step 3.1.2.2.1

Add $- 216$ and $216$ .

$f (6) = 0 + 648$

Step 3.1.2.2.2

Add $0$ and $648$ .

$f (6) = 648$

Step 3.1.2.3

The final answer is $648$ .

$648$

Step 3.2

The point found by substituting $6$ in $f (x) = - \frac{1}{6} x^{4} + x^{3} + 18 x^{2}$ is $(6, 648)$ . This point can be an inflection point.

$(6, 648)$

Step 3.3

Substitute $- 3$ in $f (x) = - \frac{1}{6} x^{4} + x^{3} + 18 x^{2}$ to find the value of $y$ .

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Step 3.3.1

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

$f (- 3) = - \frac{1}{6} \cdot {(- 3)}^{4} + {(- 3)}^{3} + 18 {(- 3)}^{2}$

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

$f (- 3) = - \frac{1}{6} \cdot 81 + {(- 3)}^{3} + 18 {(- 3)}^{2}$

Step 3.3.2.1.2

Cancel the common factor of $3$ .

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Step 3.3.2.1.2.1

Move the leading negative in $- \frac{1}{6}$ into the numerator.

$f (- 3) = \frac{- 1}{6} \cdot 81 + {(- 3)}^{3} + 18 {(- 3)}^{2}$

Step 3.3.2.1.2.2

Factor $3$ out of $6$ .

$f (- 3) = \frac{- 1}{3 (2)} \cdot 81 + {(- 3)}^{3} + 18 {(- 3)}^{2}$

Step 3.3.2.1.2.3

Factor $3$ out of $81$ .

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

Step 3.3.2.1.2.4

Cancel the common factor.

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

Step 3.3.2.1.2.5

Rewrite the expression.

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

Step 3.3.2.1.3

Combine $\frac{- 1}{2}$ and $27$ .

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

Step 3.3.2.1.4

Multiply $- 1$ by $27$ .

$f (- 3) = \frac{- 27}{2} + {(- 3)}^{3} + 18 {(- 3)}^{2}$

Step 3.3.2.1.5

Move the negative in front of the fraction.

$f (- 3) = - \frac{27}{2} + {(- 3)}^{3} + 18 {(- 3)}^{2}$

Step 3.3.2.1.6

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

$f (- 3) = - \frac{27}{2} - 27 + 18 {(- 3)}^{2}$

Step 3.3.2.1.7

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

$f (- 3) = - \frac{27}{2} - 27 + 18 \cdot 9$

Step 3.3.2.1.8

Multiply $18$ by $9$ .

$f (- 3) = - \frac{27}{2} - 27 + 162$

Step 3.3.2.2

Find the common denominator.

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Step 3.3.2.2.1

Write $- 27$ as a fraction with denominator $1$ .

$f (- 3) = - \frac{27}{2} + \frac{- 27}{1} + 162$

Step 3.3.2.2.2

Multiply $\frac{- 27}{1}$ by $\frac{2}{2}$ .

$f (- 3) = - \frac{27}{2} + \frac{- 27}{1} \cdot \frac{2}{2} + 162$

Step 3.3.2.2.3

Multiply $\frac{- 27}{1}$ by $\frac{2}{2}$ .

$f (- 3) = - \frac{27}{2} + \frac{- 27 \cdot 2}{2} + 162$

Step 3.3.2.2.4

Write $162$ as a fraction with denominator $1$ .

$f (- 3) = - \frac{27}{2} + \frac{- 27 \cdot 2}{2} + \frac{162}{1}$

Step 3.3.2.2.5

Multiply $\frac{162}{1}$ by $\frac{2}{2}$ .

$f (- 3) = - \frac{27}{2} + \frac{- 27 \cdot 2}{2} + \frac{162}{1} \cdot \frac{2}{2}$

Step 3.3.2.2.6

Multiply $\frac{162}{1}$ by $\frac{2}{2}$ .

$f (- 3) = - \frac{27}{2} + \frac{- 27 \cdot 2}{2} + \frac{162 \cdot 2}{2}$

Step 3.3.2.3

Combine the numerators over the common denominator.

$f (- 3) = \frac{- 27 - 27 \cdot 2 + 162 \cdot 2}{2}$

Step 3.3.2.4

Simplify each term.

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Step 3.3.2.4.1

Multiply $- 27$ by $2$ .

$f (- 3) = \frac{- 27 - 54 + 162 \cdot 2}{2}$

Step 3.3.2.4.2

Multiply $162$ by $2$ .

$f (- 3) = \frac{- 27 - 54 + 324}{2}$

Step 3.3.2.5

Simplify by adding and subtracting.

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Step 3.3.2.5.1

Subtract $54$ from $- 27$ .

$f (- 3) = \frac{- 81 + 324}{2}$

Step 3.3.2.5.2

Add $- 81$ and $324$ .

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

Step 3.3.2.6

The final answer is $\frac{243}{2}$ .

$\frac{243}{2}$

Step 3.4

The point found by substituting $- 3$ in $f (x) = - \frac{1}{6} x^{4} + x^{3} + 18 x^{2}$ is $(- 3, \frac{243}{2})$ . This point can be an inflection point.

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

Step 3.5

Determine the points that could be inflection points.

$(6, 648), (- 3, \frac{243}{2})$

Step 4

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

$(- \infty, - 3) \cup (- 3, 6) \cup (6, \infty)$

Step 5

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

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Step 5.1

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

$f'' (- 3.1) = - 2 {(- 3.1)}^{2} + 6 (- 3.1) + 36$

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

$f'' (- 3.1) = - 2 \cdot 9.61 + 6 (- 3.1) + 36$

Step 5.2.1.2

Multiply $- 2$ by $9.61$ .

$f'' (- 3.1) = - 19.22 + 6 (- 3.1) + 36$

Step 5.2.1.3

Multiply $6$ by $- 3.1$ .

$f'' (- 3.1) = - 19.22 - 18.6 + 36$

Step 5.2.2

Simplify by adding and subtracting.

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Step 5.2.2.1

Subtract $18.6$ from $- 19.22$ .

$f'' (- 3.1) = - 37.82 + 36$

Step 5.2.2.2

Add $- 37.82$ and $36$ .

$f'' (- 3.1) = - 1.82$

Step 5.2.3

The final answer is $- 1.82$ .

$- 1.82$

Step 5.3

At $- 3.1$ , the second derivative is $- 1.82$ . Since this is negative, the second derivative is decreasing on the interval $(- \infty, - 3)$

Decreasing on $(- \infty, - 3)$ since $f'' (x) < 0$

Step 6

Substitute a value from the interval $(- 3, 6)$ into the second derivative to determine if it is increasing or decreasing.

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Step 6.1

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

$f'' (\frac{3}{2}) = - 2 {(\frac{3}{2})}^{2} + 6 (\frac{3}{2}) + 36$

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

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

$f'' (\frac{3}{2}) = - 2 \frac{3^{2}}{2^{2}} + 6 (\frac{3}{2}) + 36$

Step 6.2.1.2

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

$f'' (\frac{3}{2}) = - 2 \frac{9}{2^{2}} + 6 (\frac{3}{2}) + 36$

Step 6.2.1.3

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

$f'' (\frac{3}{2}) = - 2 (\frac{9}{4}) + 6 (\frac{3}{2}) + 36$

Step 6.2.1.4

Cancel the common factor of $2$ .

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Step 6.2.1.4.1

Factor $2$ out of $- 2$ .

$f'' (\frac{3}{2}) = 2 (- 1) (\frac{9}{4}) + 6 (\frac{3}{2}) + 36$

Step 6.2.1.4.2

Factor $2$ out of $4$ .

$f'' (\frac{3}{2}) = 2 \cdot (- 1 \frac{9}{2 \cdot 2}) + 6 (\frac{3}{2}) + 36$

Step 6.2.1.4.3

Cancel the common factor.

$f'' (\frac{3}{2}) = 2 \cdot (- 1 \frac{9}{2 \cdot 2}) + 6 (\frac{3}{2}) + 36$

Step 6.2.1.4.4

Rewrite the expression.

$f'' (\frac{3}{2}) = - 1 (\frac{9}{2}) + 6 (\frac{3}{2}) + 36$

Step 6.2.1.5

Rewrite $- 1 (\frac{9}{2})$ as $- (\frac{9}{2})$ .

$f'' (\frac{3}{2}) = - (\frac{9}{2}) + 6 (\frac{3}{2}) + 36$

Step 6.2.1.6

Cancel the common factor of $2$ .

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Step 6.2.1.6.1

Factor $2$ out of $6$ .

$f'' (\frac{3}{2}) = - \frac{9}{2} + 2 (3) (\frac{3}{2}) + 36$

Step 6.2.1.6.2

Cancel the common factor.

$f'' (\frac{3}{2}) = - \frac{9}{2} + 2 \cdot (3 (\frac{3}{2})) + 36$

Step 6.2.1.6.3

Rewrite the expression.

$f'' (\frac{3}{2}) = - \frac{9}{2} + 3 \cdot 3 + 36$

Step 6.2.1.7

Multiply $3$ by $3$ .

$f'' (\frac{3}{2}) = - \frac{9}{2} + 9 + 36$

Step 6.2.2

Find the common denominator.

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Step 6.2.2.1

Write $9$ as a fraction with denominator $1$ .

$f'' (\frac{3}{2}) = - \frac{9}{2} + \frac{9}{1} + 36$

Step 6.2.2.2

Multiply $\frac{9}{1}$ by $\frac{2}{2}$ .

$f'' (\frac{3}{2}) = - \frac{9}{2} + \frac{9}{1} \cdot \frac{2}{2} + 36$

Step 6.2.2.3

Multiply $\frac{9}{1}$ by $\frac{2}{2}$ .

$f'' (\frac{3}{2}) = - \frac{9}{2} + \frac{9 \cdot 2}{2} + 36$

Step 6.2.2.4

Write $36$ as a fraction with denominator $1$ .

$f'' (\frac{3}{2}) = - \frac{9}{2} + \frac{9 \cdot 2}{2} + \frac{36}{1}$

Step 6.2.2.5

Multiply $\frac{36}{1}$ by $\frac{2}{2}$ .

$f'' (\frac{3}{2}) = - \frac{9}{2} + \frac{9 \cdot 2}{2} + \frac{36}{1} \cdot \frac{2}{2}$

Step 6.2.2.6

Multiply $\frac{36}{1}$ by $\frac{2}{2}$ .

$f'' (\frac{3}{2}) = - \frac{9}{2} + \frac{9 \cdot 2}{2} + \frac{36 \cdot 2}{2}$

Step 6.2.3

Combine the numerators over the common denominator.

$f'' (\frac{3}{2}) = \frac{- 9 + 9 \cdot 2 + 36 \cdot 2}{2}$

Step 6.2.4

Simplify each term.

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Step 6.2.4.1

Multiply $9$ by $2$ .

$f'' (\frac{3}{2}) = \frac{- 9 + 18 + 36 \cdot 2}{2}$

Step 6.2.4.2

Multiply $36$ by $2$ .

$f'' (\frac{3}{2}) = \frac{- 9 + 18 + 72}{2}$

Step 6.2.5

Simplify by adding numbers.

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Step 6.2.5.1

Add $- 9$ and $18$ .

$f'' (\frac{3}{2}) = \frac{9 + 72}{2}$

Step 6.2.5.2

Add $9$ and $72$ .

$f'' (\frac{3}{2}) = \frac{81}{2}$

Step 6.2.6

The final answer is $\frac{81}{2}$ .

$\frac{81}{2}$

Step 6.3

At $\frac{3}{2}$ , the second derivative is $40.5$ . Since this is positive, the second derivative is increasing on the interval $(- 3, 6)$ .

Increasing on $(- 3, 6)$ since $f'' (x) > 0$

Step 7

Substitute a value from the interval $(6, \infty)$ into the second derivative to determine if it is increasing or decreasing.

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Step 7.1

Replace the variable $x$ with $6.1$ in the expression.

$f'' (6.1) = - 2 {(6.1)}^{2} + 6 (6.1) + 36$

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 $6.1$ to the power of $2$ .

$f'' (6.1) = - 2 \cdot 37.21 + 6 (6.1) + 36$

Step 7.2.1.2

Multiply $- 2$ by $37.21$ .

$f'' (6.1) = - 74.42 + 6 (6.1) + 36$

Step 7.2.1.3

Multiply $6$ by $6.1$ .

$f'' (6.1) = - 74.42 + 36.6 + 36$

Step 7.2.2

Simplify by adding numbers.

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Step 7.2.2.1

Add $- 74.42$ and $36.6$ .

$f'' (6.1) = - 37.82 + 36$

Step 7.2.2.2

Add $- 37.82$ and $36$ .

$f'' (6.1) = - 1.82$

Step 7.2.3

The final answer is $- 1.82$ .

$- 1.82$

Step 7.3

At $6.1$ , the second derivative is $- 1.82$ . Since this is negative, the second derivative is decreasing on the interval $(6, \infty)$

Decreasing on $(6, \infty)$ 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 $(- 3, \frac{243}{2}), (6, 648)$ .

$(- 3, \frac{243}{2}), (6, 648)$

Step 9