Calculus Examples

$f (x) = \frac{x - 4}{x^{3}}$

Step 1

Find the $x$ values where the second derivative is equal to $0$ .

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Step 1.1

Find the second derivative.

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Step 1.1.1

Find the first derivative.

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Step 1.1.1.1

Differentiate using the Quotient Rule which states that $\frac{d}{d x} [\frac{f (x)}{g (x)}]$ is $\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ where $f (x) = x - 4$ and $g (x) = x^{3}$ .

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

Step 1.1.1.2

Differentiate.

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Step 1.1.1.2.1

Multiply the exponents in ${(x^{3})}^{2}$ .

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Step 1.1.1.2.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 1.1.1.2.1.2

Multiply $3$ by $2$ .

$f' (x) = \frac{x^{3} \frac{d}{d x} (x - 4) - (x - 4) \frac{d}{d x} x^{3}}{x^{6}}$

Step 1.1.1.2.2

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

$f' (x) = \frac{x^{3} (\frac{d}{d x} (x) + \frac{d}{d x} (- 4)) - (x - 4) \frac{d}{d x} x^{3}}{x^{6}}$

Step 1.1.1.2.3

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

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

Step 1.1.1.2.4

Since $- 4$ is constant with respect to $x$ , the derivative of $- 4$ with respect to $x$ is $0$ .

$f' (x) = \frac{x^{3} (1 + 0) - (x - 4) \frac{d}{d x} x^{3}}{x^{6}}$

Step 1.1.1.2.5

Simplify the expression.

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Step 1.1.1.2.5.1

Add $1$ and $0$ .

$f' (x) = \frac{x^{3} \cdot 1 - (x - 4) \frac{d}{d x} x^{3}}{x^{6}}$

Step 1.1.1.2.5.2

Multiply $x^{3}$ by $1$ .

$f' (x) = \frac{x^{3} - (x - 4) \frac{d}{d x} x^{3}}{x^{6}}$

Step 1.1.1.2.6

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

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

Step 1.1.1.2.7

Simplify with factoring out.

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Step 1.1.1.2.7.1

Multiply $3$ by $- 1$ .

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

Step 1.1.1.2.7.2

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

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Step 1.1.1.2.7.2.1

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

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

Step 1.1.1.2.7.2.2

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

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

Step 1.1.1.2.7.2.3

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

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

Step 1.1.1.3

Cancel the common factors.

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Step 1.1.1.3.1

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

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

Step 1.1.1.3.2

Cancel the common factor.

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

Step 1.1.1.3.3

Rewrite the expression.

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

Step 1.1.1.4

Simplify.

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Step 1.1.1.4.1

Apply the distributive property.

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

Step 1.1.1.4.2

Simplify the numerator.

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Step 1.1.1.4.2.1

Multiply $- 3$ by $- 4$ .

$f' (x) = \frac{x - 3 x + 12}{x^{4}}$

Step 1.1.1.4.2.2

Subtract $3 x$ from $x$ .

$f' (x) = \frac{- 2 x + 12}{x^{4}}$

Step 1.1.1.4.3

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

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Step 1.1.1.4.3.1

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

$f' (x) = \frac{2 (- x) + 12}{x^{4}}$

Step 1.1.1.4.3.2

Factor $2$ out of $12$ .

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

Step 1.1.1.4.3.3

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

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

Step 1.1.1.4.4

Factor $- 1$ out of $- x$ .

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

Step 1.1.1.4.5

Rewrite $6$ as $- 1 (- 6)$ .

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

Step 1.1.1.4.6

Factor $- 1$ out of $- (x) - 1 (- 6)$ .

$f' (x) = \frac{2 (- (x - 6))}{x^{4}}$

Step 1.1.1.4.7

Rewrite $- (x - 6)$ as $- 1 (x - 6)$ .

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

Step 1.1.1.4.8

Move the negative in front of the fraction.

$f' (x) = - \frac{2 (x - 6)}{x^{4}}$

Step 1.1.2

Find the second derivative.

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Step 1.1.2.1

Since $- 2$ is constant with respect to $x$ , the derivative of $- \frac{2 (x - 6)}{x^{4}}$ with respect to $x$ is $- 2 \frac{d}{d x} [\frac{x - 6}{x^{4}}]$ .

$f'' (x) = - 2 \frac{d}{d x} \frac{x - 6}{x^{4}}$

Step 1.1.2.2

$f'' (x) = - 2 \frac{x^{4} \frac{d}{d x} (x - 6) - (x - 6) \frac{d}{d x} x^{4}}{{(x^{4})}^{2}}$

Step 1.1.2.3

Differentiate.

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Step 1.1.2.3.1

Multiply the exponents in ${(x^{4})}^{2}$ .

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Step 1.1.2.3.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$f'' (x) = - 2 \frac{x^{4} \frac{d}{d x} (x - 6) - (x - 6) \frac{d}{d x} x^{4}}{x^{4 \cdot 2}}$

Step 1.1.2.3.1.2

Multiply $4$ by $2$ .

$f'' (x) = - 2 \frac{x^{4} \frac{d}{d x} (x - 6) - (x - 6) \frac{d}{d x} x^{4}}{x^{8}}$

Step 1.1.2.3.2

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

$f'' (x) = - 2 \frac{x^{4} (\frac{d}{d x} (x) + \frac{d}{d x} (- 6)) - (x - 6) \frac{d}{d x} x^{4}}{x^{8}}$

Step 1.1.2.3.3

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

$f'' (x) = - 2 \frac{x^{4} (1 + \frac{d}{d x} (- 6)) - (x - 6) \frac{d}{d x} x^{4}}{x^{8}}$

Step 1.1.2.3.4

Since $- 6$ is constant with respect to $x$ , the derivative of $- 6$ with respect to $x$ is $0$ .

$f'' (x) = - 2 \frac{x^{4} (1 + 0) - (x - 6) \frac{d}{d x} x^{4}}{x^{8}}$

Step 1.1.2.3.5

Simplify the expression.

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Step 1.1.2.3.5.1

Add $1$ and $0$ .

$f'' (x) = - 2 \frac{x^{4} \cdot 1 - (x - 6) \frac{d}{d x} x^{4}}{x^{8}}$

Step 1.1.2.3.5.2

Multiply $x^{4}$ by $1$ .

$f'' (x) = - 2 \frac{x^{4} - (x - 6) \frac{d}{d x} x^{4}}{x^{8}}$

Step 1.1.2.3.6

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

$f'' (x) = - 2 \frac{x^{4} - (x - 6) (4 x^{3})}{x^{8}}$

Step 1.1.2.3.7

Simplify with factoring out.

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Step 1.1.2.3.7.1

Multiply $4$ by $- 1$ .

$f'' (x) = - 2 \frac{x^{4} - 4 (x - 6) x^{3}}{x^{8}}$

Step 1.1.2.3.7.2

Factor $x^{3}$ out of $x^{4} - 4 (x - 6) x^{3}$ .

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Step 1.1.2.3.7.2.1

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

$f'' (x) = - 2 \frac{x^{3} x - 4 (x - 6) x^{3}}{x^{8}}$

Step 1.1.2.3.7.2.2

Factor $x^{3}$ out of $- 4 (x - 6) x^{3}$ .

$f'' (x) = - 2 \frac{x^{3} x + x^{3} (- 4 (x - 6))}{x^{8}}$

Step 1.1.2.3.7.2.3

Factor $x^{3}$ out of $x^{3} x + x^{3} (- 4 (x - 6))$ .

$f'' (x) = - 2 \frac{x^{3} (x - 4 (x - 6))}{x^{8}}$

Step 1.1.2.4

Cancel the common factors.

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Step 1.1.2.4.1

Factor $x^{3}$ out of $x^{8}$ .

$f'' (x) = - 2 \frac{x^{3} (x - 4 (x - 6))}{x^{3} x^{5}}$

Step 1.1.2.4.2

Cancel the common factor.

$f'' (x) = - 2 \frac{x^{3} (x - 4 (x - 6))}{x^{3} x^{5}}$

Step 1.1.2.4.3

Rewrite the expression.

$f'' (x) = - 2 \frac{x - 4 (x - 6)}{x^{5}}$

Step 1.1.2.5

Combine $- 2$ and $\frac{x - 4 (x - 6)}{x^{5}}$ .

$f'' (x) = \frac{- 2 (x - 4 (x - 6))}{x^{5}}$

Step 1.1.2.6

Move the negative in front of the fraction.

$f'' (x) = - \frac{(2) (x - 4 (x - 6))}{x^{5}}$

Step 1.1.2.7

Simplify.

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Step 1.1.2.7.1

Apply the distributive property.

$f'' (x) = - \frac{2 (x - 4 x - 4 \cdot - 6)}{x^{5}}$

Step 1.1.2.7.2

Apply the distributive property.

$f'' (x) = - \frac{2 x + 2 (- 4 x) + 2 (- 4 \cdot - 6)}{x^{5}}$

Step 1.1.2.7.3

Simplify the numerator.

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Step 1.1.2.7.3.1

Simplify each term.

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Step 1.1.2.7.3.1.1

Multiply $- 4$ by $2$ .

$f'' (x) = - \frac{2 x - 8 x + 2 (- 4 \cdot - 6)}{x^{5}}$

Step 1.1.2.7.3.1.2

Multiply $2 (- 4 \cdot - 6)$ .

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Step 1.1.2.7.3.1.2.1

Multiply $- 4$ by $- 6$ .

$f'' (x) = - \frac{2 x - 8 x + 2 \cdot 24}{x^{5}}$

Step 1.1.2.7.3.1.2.2

Multiply $2$ by $24$ .

$f'' (x) = - \frac{2 x - 8 x + 48}{x^{5}}$

Step 1.1.2.7.3.2

Subtract $8 x$ from $2 x$ .

$f'' (x) = - \frac{- 6 x + 48}{x^{5}}$

Step 1.1.2.7.4

Factor $6$ out of $- 6 x + 48$ .

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Step 1.1.2.7.4.1

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

$f'' (x) = - \frac{6 (- x) + 48}{x^{5}}$

Step 1.1.2.7.4.2

Factor $6$ out of $48$ .

$f'' (x) = - \frac{6 (- x) + 6 (8)}{x^{5}}$

Step 1.1.2.7.4.3

Factor $6$ out of $6 (- x) + 6 (8)$ .

$f'' (x) = - \frac{6 (- x + 8)}{x^{5}}$

Step 1.1.2.7.5

Factor $- 1$ out of $- x$ .

$f'' (x) = - \frac{6 (- (x) + 8)}{x^{5}}$

Step 1.1.2.7.6

Rewrite $8$ as $- 1 (- 8)$ .

$f'' (x) = - \frac{6 (- (x) - 1 \cdot - 8)}{x^{5}}$

Step 1.1.2.7.7

Factor $- 1$ out of $- (x) - 1 (- 8)$ .

$f'' (x) = - \frac{6 (- (x - 8))}{x^{5}}$

Step 1.1.2.7.8

Rewrite $- (x - 8)$ as $- 1 (x - 8)$ .

$f'' (x) = - \frac{6 (- 1 (x - 8))}{x^{5}}$

Step 1.1.2.7.9

Move the negative in front of the fraction.

$f'' (x) = \frac{6 (x - 8)}{x^{5}}$

Step 1.1.2.7.10

Multiply $- 1$ by $- 1$ .

$f'' (x) = 1 (\frac{6 (x - 8)}{x^{5}})$

Step 1.1.2.7.11

Multiply $\frac{6 (x - 8)}{x^{5}}$ by $1$ .

$f'' (x) = \frac{6 (x - 8)}{x^{5}}$

Step 1.1.3

The second derivative of $f (x)$ with respect to $x$ is $\frac{6 (x - 8)}{x^{5}}$ .

$\frac{6 (x - 8)}{x^{5}}$

Step 1.2

Set the second derivative equal to $0$ then solve the equation $\frac{6 (x - 8)}{x^{5}} = 0$ .

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Step 1.2.1

Set the second derivative equal to $0$ .

$\frac{6 (x - 8)}{x^{5}} = 0$

Step 1.2.2

Set the numerator equal to zero.

$6 (x - 8) = 0$

Step 1.2.3

Solve the equation for $x$ .

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Step 1.2.3.1

Divide each term in $6 (x - 8) = 0$ by $6$ and simplify.

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Step 1.2.3.1.1

Divide each term in $6 (x - 8) = 0$ by $6$ .

$\frac{6 (x - 8)}{6} = \frac{0}{6}$

Step 1.2.3.1.2

Simplify the left side.

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Step 1.2.3.1.2.1

Cancel the common factor of $6$ .

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Step 1.2.3.1.2.1.1

Cancel the common factor.

$\frac{6 (x - 8)}{6} = \frac{0}{6}$

Step 1.2.3.1.2.1.2

Divide $x - 8$ by $1$ .

$x - 8 = \frac{0}{6}$

Step 1.2.3.1.3

Simplify the right side.

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Step 1.2.3.1.3.1

Divide $0$ by $6$ .

$x - 8 = 0$

Step 1.2.3.2

Add $8$ to both sides of the equation.

$x = 8$

Step 2

Find the domain of $f (x) = \frac{x - 4}{x^{3}}$ .

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Step 2.1

Set the denominator in $\frac{x - 4}{x^{3}}$ equal to $0$ to find where the expression is undefined.

$x^{3} = 0$

Step 2.2

Solve for $x$ .

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Step 2.2.1

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \sqrt[3]{0}$

Step 2.2.2

Simplify $\sqrt[3]{0}$ .

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Step 2.2.2.1

Rewrite $0$ as $0^{3}$ .

$x = \sqrt[3]{0^{3}}$

Step 2.2.2.2

Pull terms out from under the radical, assuming real numbers.

$x = 0$

Step 2.3

The domain is all values of $x$ that make the expression defined.

Interval Notation:

$(- \infty, 0) \cup (0, \infty)$

Set-Builder Notation:

${x | x \neq 0}$

Interval Notation:

$(- \infty, 0) \cup (0, \infty)$

Set-Builder Notation:

${x | x \neq 0}$

Step 3

Create intervals around the $x$ -values where the second derivative is zero or undefined.

$(- \infty, 0) \cup (0, 8) \cup (8, \infty)$

Step 4

Substitute any number from the interval $(- \infty, 0)$ into the second derivative and evaluate to determine the concavity.

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Step 4.1

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

$f'' (- 2) = \frac{6 ((- 2) - 8)}{{(- 2)}^{5}}$

Step 4.2

Simplify the result.

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Step 4.2.1

Simplify the expression.

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Step 4.2.1.1

Subtract $8$ from $- 2$ .

$f'' (- 2) = \frac{6 \cdot - 10}{{(- 2)}^{5}}$

Step 4.2.1.2

Raise $- 2$ to the power of $5$ .

$f'' (- 2) = \frac{6 \cdot - 10}{- 32}$

Step 4.2.1.3

Multiply $6$ by $- 10$ .

$f'' (- 2) = \frac{- 60}{- 32}$

Step 4.2.2

Cancel the common factor of $- 60$ and $- 32$ .

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Step 4.2.2.1

Factor $- 4$ out of $- 60$ .

$f'' (- 2) = \frac{- 4 \cdot 15}{- 32}$

Step 4.2.2.2

Cancel the common factors.

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Step 4.2.2.2.1

Factor $- 4$ out of $- 32$ .

$f'' (- 2) = \frac{- 4 \cdot 15}{- 4 \cdot 8}$

Step 4.2.2.2.2

Cancel the common factor.

$f'' (- 2) = \frac{- 4 \cdot 15}{- 4 \cdot 8}$

Step 4.2.2.2.3

Rewrite the expression.

$f'' (- 2) = \frac{15}{8}$

Step 4.2.3

The final answer is $\frac{15}{8}$ .

$\frac{15}{8}$

Step 4.3

The graph is concave up on the interval $(- \infty, 0)$ because $f'' (- 2)$ is positive.

Concave up on $(- \infty, 0)$ since $f'' (x)$ is positive

Step 5

Substitute any number from the interval $(0, 8)$ into the second derivative and evaluate to determine the concavity.

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Step 5.1

Replace the variable $x$ with $4$ in the expression.

$f'' (4) = \frac{6 ((4) - 8)}{{(4)}^{5}}$

Step 5.2

Simplify the result.

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Step 5.2.1

Simplify the expression.

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Step 5.2.1.1

Subtract $8$ from $4$ .

$f'' (4) = \frac{6 \cdot - 4}{4^{5}}$

Step 5.2.1.2

Raise $4$ to the power of $5$ .

$f'' (4) = \frac{6 \cdot - 4}{1024}$

Step 5.2.1.3

Multiply $6$ by $- 4$ .

$f'' (4) = \frac{- 24}{1024}$

Step 5.2.2

Cancel the common factor of $- 24$ and $1024$ .

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Step 5.2.2.1

Factor $8$ out of $- 24$ .

$f'' (4) = \frac{8 (- 3)}{1024}$

Step 5.2.2.2

Cancel the common factors.

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Step 5.2.2.2.1

Factor $8$ out of $1024$ .

$f'' (4) = \frac{8 \cdot - 3}{8 \cdot 128}$

Step 5.2.2.2.2

Cancel the common factor.

$f'' (4) = \frac{8 \cdot - 3}{8 \cdot 128}$

Step 5.2.2.2.3

Rewrite the expression.

$f'' (4) = \frac{- 3}{128}$

Step 5.2.3

Move the negative in front of the fraction.

$f'' (4) = - \frac{3}{128}$

Step 5.2.4

The final answer is $- \frac{3}{128}$ .

$- \frac{3}{128}$

Step 5.3

The graph is concave down on the interval $(0, 8)$ because $f'' (4)$ is negative.

Concave down on $(0, 8)$ since $f'' (x)$ is negative

Step 6

Substitute any number from the interval $(8, \infty)$ into the second derivative and evaluate to determine the concavity.

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Step 6.1

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

$f'' (10) = \frac{6 ((10) - 8)}{{(10)}^{5}}$

Step 6.2

Simplify the result.

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Step 6.2.1

Simplify the expression.

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Step 6.2.1.1

Subtract $8$ from $10$ .

$f'' (10) = \frac{6 \cdot 2}{10^{5}}$

Step 6.2.1.2

Raise $10$ to the power of $5$ .

$f'' (10) = \frac{6 \cdot 2}{100000}$

Step 6.2.1.3

Multiply $6$ by $2$ .

$f'' (10) = \frac{12}{100000}$

Step 6.2.2

Cancel the common factor of $12$ and $100000$ .

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Step 6.2.2.1

Factor $4$ out of $12$ .

$f'' (10) = \frac{4 (3)}{100000}$

Step 6.2.2.2

Cancel the common factors.

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Step 6.2.2.2.1

Factor $4$ out of $100000$ .

$f'' (10) = \frac{4 \cdot 3}{4 \cdot 25000}$

Step 6.2.2.2.2

Cancel the common factor.

$f'' (10) = \frac{4 \cdot 3}{4 \cdot 25000}$

Step 6.2.2.2.3

Rewrite the expression.

$f'' (10) = \frac{3}{25000}$

Step 6.2.3

The final answer is $\frac{3}{25000}$ .

$\frac{3}{25000}$

Step 6.3

The graph is concave up on the interval $(8, \infty)$ because $f'' (10)$ is positive.

Concave up on $(8, \infty)$ since $f'' (x)$ is positive

Step 7

The graph is concave down when the second derivative is negative and concave up when the second derivative is positive.

Concave up on $(- \infty, 0)$ since $f'' (x)$ is positive

Concave down on $(0, 8)$ since $f'' (x)$ is negative

Concave up on $(8, \infty)$ since $f'' (x)$ is positive

Step 8