Calculus Examples

$f (x) = 64 x^{2} + \frac{128}{x} + 3$

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

By the Sum Rule, the derivative of $64 x^{2} + \frac{128}{x} + 3$ with respect to $x$ is $\frac{d}{d x} [64 x^{2}] + \frac{d}{d x} [\frac{128}{x}] + \frac{d}{d x} [3]$ .

$\frac{d}{d x} [64 x^{2}] + \frac{d}{d x} [\frac{128}{x}] + \frac{d}{d x} [3]$

Step 1.1.1.2

Evaluate $\frac{d}{d x} [64 x^{2}]$ .

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Step 1.1.1.2.1

Since $64$ is constant with respect to $x$ , the derivative of $64 x^{2}$ with respect to $x$ is $64 \frac{d}{d x} [x^{2}]$ .

$64 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [\frac{128}{x}] + \frac{d}{d x} [3]$

Step 1.1.1.2.2

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

$64 (2 x) + \frac{d}{d x} [\frac{128}{x}] + \frac{d}{d x} [3]$

Step 1.1.1.2.3

Multiply $2$ by $64$ .

$128 x + \frac{d}{d x} [\frac{128}{x}] + \frac{d}{d x} [3]$

Step 1.1.1.3

Evaluate $\frac{d}{d x} [\frac{128}{x}]$ .

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Step 1.1.1.3.1

Since $128$ is constant with respect to $x$ , the derivative of $\frac{128}{x}$ with respect to $x$ is $128 \frac{d}{d x} [\frac{1}{x}]$ .

$128 x + 128 \frac{d}{d x} [\frac{1}{x}] + \frac{d}{d x} [3]$

Step 1.1.1.3.2

Rewrite $\frac{1}{x}$ as $x^{- 1}$ .

$128 x + 128 \frac{d}{d x} [x^{- 1}] + \frac{d}{d x} [3]$

Step 1.1.1.3.3

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

$128 x + 128 (- x^{- 2}) + \frac{d}{d x} [3]$

Step 1.1.1.3.4

Multiply $- 1$ by $128$ .

$128 x - 128 x^{- 2} + \frac{d}{d x} [3]$

Step 1.1.1.4

Since $3$ is constant with respect to $x$ , the derivative of $3$ with respect to $x$ is $0$ .

$128 x - 128 x^{- 2} + 0$

Step 1.1.1.5

Simplify.

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Step 1.1.1.5.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$128 x - 128 \frac{1}{x^{2}} + 0$

Step 1.1.1.5.2

Combine terms.

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Step 1.1.1.5.2.1

Combine $- 128$ and $\frac{1}{x^{2}}$ .

$128 x + \frac{- 128}{x^{2}} + 0$

Step 1.1.1.5.2.2

Move the negative in front of the fraction.

$128 x - \frac{128}{x^{2}} + 0$

Step 1.1.1.5.2.3

Add $128 x - \frac{128}{x^{2}}$ and $0$ .

$f' (x) = 128 x - \frac{128}{x^{2}}$

Step 1.1.2

Find the second derivative.

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Step 1.1.2.1

By the Sum Rule, the derivative of $128 x - \frac{128}{x^{2}}$ with respect to $x$ is $\frac{d}{d x} [128 x] + \frac{d}{d x} [- \frac{128}{x^{2}}]$ .

$\frac{d}{d x} [128 x] + \frac{d}{d x} [- \frac{128}{x^{2}}]$

Step 1.1.2.2

Evaluate $\frac{d}{d x} [128 x]$ .

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Step 1.1.2.2.1

Since $128$ is constant with respect to $x$ , the derivative of $128 x$ with respect to $x$ is $128 \frac{d}{d x} [x]$ .

$128 \frac{d}{d x} [x] + \frac{d}{d x} [- \frac{128}{x^{2}}]$

Step 1.1.2.2.2

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

$128 \cdot 1 + \frac{d}{d x} [- \frac{128}{x^{2}}]$

Step 1.1.2.2.3

Multiply $128$ by $1$ .

$128 + \frac{d}{d x} [- \frac{128}{x^{2}}]$

Step 1.1.2.3

Evaluate $\frac{d}{d x} [- \frac{128}{x^{2}}]$ .

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Step 1.1.2.3.1

Since $- 128$ is constant with respect to $x$ , the derivative of $- \frac{128}{x^{2}}$ with respect to $x$ is $- 128 \frac{d}{d x} [\frac{1}{x^{2}}]$ .

$128 - 128 \frac{d}{d x} [\frac{1}{x^{2}}]$

Step 1.1.2.3.2

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

$128 - 128 \frac{d}{d x} [{(x^{2})}^{- 1}]$

Step 1.1.2.3.3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{- 1}$ and $g (x) = x^{2}$ .

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Step 1.1.2.3.3.1

To apply the Chain Rule, set $u$ as $x^{2}$ .

$128 - 128 (\frac{d}{d u} [u^{- 1}] \frac{d}{d x} [x^{2}])$

Step 1.1.2.3.3.2

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

$128 - 128 (- u^{- 2} \frac{d}{d x} [x^{2}])$

Step 1.1.2.3.3.3

Replace all occurrences of $u$ with $x^{2}$ .

$128 - 128 (- {(x^{2})}^{- 2} \frac{d}{d x} [x^{2}])$

Step 1.1.2.3.4

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

$128 - 128 (- {(x^{2})}^{- 2} (2 x))$

Step 1.1.2.3.5

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

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Step 1.1.2.3.5.1

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

$128 - 128 (- x^{2 \cdot - 2} (2 x))$

Step 1.1.2.3.5.2

Multiply $2$ by $- 2$ .

$128 - 128 (- x^{- 4} (2 x))$

Step 1.1.2.3.6

Multiply $2$ by $- 1$ .

$128 - 128 (- 2 x^{- 4} x)$

Step 1.1.2.3.7

Raise $x$ to the power of $1$ .

$128 - 128 (- 2 (x^{1} x^{- 4}))$

Step 1.1.2.3.8

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$128 - 128 (- 2 x^{1 - 4})$

Step 1.1.2.3.9

Subtract $4$ from $1$ .

$128 - 128 (- 2 x^{- 3})$

Step 1.1.2.3.10

Multiply $- 2$ by $- 128$ .

$128 + 256 x^{- 3}$

Step 1.1.2.4

Simplify.

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Step 1.1.2.4.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$128 + 256 \frac{1}{x^{3}}$

Step 1.1.2.4.2

Combine $256$ and $\frac{1}{x^{3}}$ .

$128 + \frac{256}{x^{3}}$

Step 1.1.2.4.3

Reorder terms.

$f'' (x) = \frac{256}{x^{3}} + 128$

Step 1.1.3

The second derivative of $f (x)$ with respect to $x$ is $\frac{256}{x^{3}} + 128$ .

$\frac{256}{x^{3}} + 128$

Step 1.2

Set the second derivative equal to $0$ then solve the equation $\frac{256}{x^{3}} + 128 = 0$ .

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Step 1.2.1

Set the second derivative equal to $0$ .

$\frac{256}{x^{3}} + 128 = 0$

Step 1.2.2

Subtract $128$ from both sides of the equation.

$\frac{256}{x^{3}} = - 128$

Step 1.2.3

Find the LCD of the terms in the equation.

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Step 1.2.3.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$x^{3}, 1$

Step 1.2.3.2

The LCM of one and any expression is the expression.

$x^{3}$

Step 1.2.4

Multiply each term in $\frac{256}{x^{3}} = - 128$ by $x^{3}$ to eliminate the fractions.

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Step 1.2.4.1

Multiply each term in $\frac{256}{x^{3}} = - 128$ by $x^{3}$ .

$\frac{256}{x^{3}} x^{3} = - 128 x^{3}$

Step 1.2.4.2

Simplify the left side.

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Step 1.2.4.2.1

Cancel the common factor of $x^{3}$ .

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Step 1.2.4.2.1.1

Cancel the common factor.

$\frac{256}{x^{3}} x^{3} = - 128 x^{3}$

Step 1.2.4.2.1.2

Rewrite the expression.

$256 = - 128 x^{3}$

Step 1.2.5

Solve the equation.

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Step 1.2.5.1

Rewrite the equation as $- 128 x^{3} = 256$ .

$- 128 x^{3} = 256$

Step 1.2.5.2

Subtract $256$ from both sides of the equation.

$- 128 x^{3} - 256 = 0$

Step 1.2.5.3

Factor $- 128$ out of $- 128 x^{3} - 256$ .

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Step 1.2.5.3.1

Factor $- 128$ out of $- 128 x^{3}$ .

$- 128 x^{3} - 256 = 0$

Step 1.2.5.3.2

Factor $- 128$ out of $- 256$ .

$- 128 x^{3} - 128 \cdot 2 = 0$

Step 1.2.5.3.3

Factor $- 128$ out of $- 128 (x^{3}) - 128 (2)$ .

$- 128 (x^{3} + 2) = 0$

Step 1.2.5.4

Divide each term in $- 128 (x^{3} + 2) = 0$ by $- 128$ and simplify.

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Step 1.2.5.4.1

Divide each term in $- 128 (x^{3} + 2) = 0$ by $- 128$ .

$\frac{- 128 (x^{3} + 2)}{- 128} = \frac{0}{- 128}$

Step 1.2.5.4.2

Simplify the left side.

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Step 1.2.5.4.2.1

Cancel the common factor of $- 128$ .

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Step 1.2.5.4.2.1.1

Cancel the common factor.

$\frac{- 128 (x^{3} + 2)}{- 128} = \frac{0}{- 128}$

Step 1.2.5.4.2.1.2

Divide $x^{3} + 2$ by $1$ .

$x^{3} + 2 = \frac{0}{- 128}$

Step 1.2.5.4.3

Simplify the right side.

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Step 1.2.5.4.3.1

Divide $0$ by $- 128$ .

$x^{3} + 2 = 0$

Step 1.2.5.5

Subtract $2$ from both sides of the equation.

$x^{3} = - 2$

Step 1.2.5.6

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

$x = \sqrt[3]{- 2}$

Step 1.2.5.7

Simplify $\sqrt[3]{- 2}$ .

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Step 1.2.5.7.1

Rewrite $- 2$ as ${(- 1)}^{3} \cdot 2$ .

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Step 1.2.5.7.1.1

Rewrite $- 2$ as $- 1 (2)$ .

$x = \sqrt[3]{- 1 (2)}$

Step 1.2.5.7.1.2

Rewrite $- 1$ as ${(- 1)}^{3}$ .

$x = \sqrt[3]{{(- 1)}^{3} \cdot 2}$

Step 1.2.5.7.2

Pull terms out from under the radical.

$x = - 1 \sqrt[3]{2}$

Step 1.2.5.7.3

Rewrite $- 1 \sqrt[3]{2}$ as $- \sqrt[3]{2}$ .

$x = - \sqrt[3]{2}$

Step 2

Find the domain of $f (x) = 64 x^{2} + \frac{128}{x} + 3$ .

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Step 2.1

Set the denominator in $\frac{128}{x}$ equal to $0$ to find where the expression is undefined.

$x = 0$

Step 2.2

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.

$(- \infty, - \sqrt[3]{2}) \cup (- \sqrt[3]{2}, 0) \cup (0, \infty)$

Step 4

Substitute any number from the interval $(- \infty, - \sqrt[3]{2})$ into the second derivative and evaluate to determine the concavity.

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Step 4.1

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

$f'' (- 4) = \frac{256}{{(- 4)}^{3}} + 128$

Step 4.2

Simplify the result.

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Step 4.2.1

Simplify each term.

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Step 4.2.1.1

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

$f'' (- 4) = \frac{256}{- 64} + 128$

Step 4.2.1.2

Divide $256$ by $- 64$ .

$f'' (- 4) = - 4 + 128$

Step 4.2.2

Add $- 4$ and $128$ .

$f'' (- 4) = 124$

Step 4.2.3

The final answer is $124$ .

$124$

Step 4.3

The graph is concave up on the interval $(- \infty, - \sqrt[3]{2})$ because $f'' (- 4)$ is positive.

Concave up on $(- \infty, - \sqrt[3]{2})$ since $f'' (x)$ is positive

Step 5

Substitute any number from the interval $(- \sqrt[3]{2}, 0)$ into the second derivative and evaluate to determine the concavity.

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Step 5.1

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

$f'' (- 1) = \frac{256}{{(- 1)}^{3}} + 128$

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

$f'' (- 1) = \frac{256}{- 1} + 128$

Step 5.2.1.2

Divide $256$ by $- 1$ .

$f'' (- 1) = - 256 + 128$

Step 5.2.2

Add $- 256$ and $128$ .

$f'' (- 1) = - 128$

Step 5.2.3

The final answer is $- 128$ .

$- 128$

Step 5.3

The graph is concave down on the interval $(- \sqrt[3]{2}, 0)$ because $f'' (- 1)$ is negative.

Concave down on $(- \sqrt[3]{2}, 0)$ since $f'' (x)$ is negative

Step 6

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

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Step 6.1

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

$f'' (2) = \frac{256}{{(2)}^{3}} + 128$

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 $2$ to the power of $3$ .

$f'' (2) = \frac{256}{8} + 128$

Step 6.2.1.2

Divide $256$ by $8$ .

$f'' (2) = 32 + 128$

Step 6.2.2

Add $32$ and $128$ .

$f'' (2) = 160$

Step 6.2.3

The final answer is $160$ .

$160$

Step 6.3

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

Concave up on $(0, \infty)$ 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 $(- \infty, - \sqrt[3]{2})$ since $f'' (x)$ is positive

Concave down on $(- \sqrt[3]{2}, 0)$ since $f'' (x)$ is negative

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

Step 8