Calculus Examples

$f (x) = x^{4} + x + 2$

Step 1

Find the first derivative of the function.

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Step 1.1

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

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

Step 1.2

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

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

Step 1.3

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

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

Step 1.4

Since $2$ is constant with respect to $x$ , the derivative of $2$ with respect to $x$ is $0$ .

$4 x^{3} + 1 + 0$

Step 1.5

Add $4 x^{3} + 1$ and $0$ .

$4 x^{3} + 1$

Step 2

Find the second derivative of the function.

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Step 2.1

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

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

Step 2.2

Evaluate $\frac{d}{d x} [4 x^{3}]$ .

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Step 2.2.1

Since $4$ is constant with respect to $x$ , the derivative of $4 x^{3}$ with respect to $x$ is $4 \frac{d}{d x} [x^{3}]$ .

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

Step 2.2.2

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

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

Step 2.2.3

Multiply $3$ by $4$ .

$f'' (x) = 12 x^{2} + \frac{d}{d x} (1)$

Step 2.3

Differentiate using the Constant Rule.

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Step 2.3.1

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$f'' (x) = 12 x^{2} + 0$

Step 2.3.2

Add $12 x^{2}$ and $0$ .

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

Step 3

To find the local maximum and minimum values of the function, set the derivative equal to $0$ and solve.

$4 x^{3} + 1 = 0$

Step 4

Find the first derivative.

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Step 4.1

Find the first derivative.

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Step 4.1.1

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

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

Step 4.1.2

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

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

Step 4.1.3

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

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

Step 4.1.4

Since $2$ is constant with respect to $x$ , the derivative of $2$ with respect to $x$ is $0$ .

$4 x^{3} + 1 + 0$

Step 4.1.5

Add $4 x^{3} + 1$ and $0$ .

$f' (x) = 4 x^{3} + 1$

Step 4.2

The first derivative of $f (x)$ with respect to $x$ is $4 x^{3} + 1$ .

$4 x^{3} + 1$

Step 5

Set the first derivative equal to $0$ then solve the equation $4 x^{3} + 1 = 0$ .

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Step 5.1

Set the first derivative equal to $0$ .

$4 x^{3} + 1 = 0$

Step 5.2

Subtract $1$ from both sides of the equation.

$4 x^{3} = - 1$

Step 5.3

Divide each term in $4 x^{3} = - 1$ by $4$ and simplify.

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Step 5.3.1

Divide each term in $4 x^{3} = - 1$ by $4$ .

$\frac{4 x^{3}}{4} = \frac{- 1}{4}$

Step 5.3.2

Simplify the left side.

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Step 5.3.2.1

Cancel the common factor of $4$ .

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Step 5.3.2.1.1

Cancel the common factor.

$\frac{4 x^{3}}{4} = \frac{- 1}{4}$

Step 5.3.2.1.2

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

$x^{3} = \frac{- 1}{4}$

Step 5.3.3

Simplify the right side.

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Step 5.3.3.1

Move the negative in front of the fraction.

$x^{3} = - \frac{1}{4}$

Step 5.4

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

$x = \sqrt[3]{- \frac{1}{4}}$

Step 5.5

Simplify $\sqrt[3]{- \frac{1}{4}}$ .

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Step 5.5.1

Rewrite $- \frac{1}{4}$ as ${({(- 1)}^{3})}^{3} \frac{1}{4}$ .

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Step 5.5.1.1

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

$x = \sqrt[3]{{(- 1)}^{3} \frac{1}{4}}$

Step 5.5.1.2

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

$x = \sqrt[3]{{({(- 1)}^{3})}^{3} \frac{1}{4}}$

Step 5.5.2

Pull terms out from under the radical.

$x = {(- 1)}^{3} \sqrt[3]{\frac{1}{4}}$

Step 5.5.3

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

$x = - \sqrt[3]{\frac{1}{4}}$

Step 5.5.4

Rewrite $\sqrt[3]{\frac{1}{4}}$ as $\frac{\sqrt[3]{1}}{\sqrt[3]{4}}$ .

$x = - \frac{\sqrt[3]{1}}{\sqrt[3]{4}}$

Step 5.5.5

Any root of $1$ is $1$ .

$x = - \frac{1}{\sqrt[3]{4}}$

Step 5.5.6

Multiply $\frac{1}{\sqrt[3]{4}}$ by $\frac{{\sqrt[3]{4}}^{2}}{{\sqrt[3]{4}}^{2}}$ .

$x = - (\frac{1}{\sqrt[3]{4}} \cdot \frac{{\sqrt[3]{4}}^{2}}{{\sqrt[3]{4}}^{2}})$

Step 5.5.7

Combine and simplify the denominator.

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Step 5.5.7.1

Multiply $\frac{1}{\sqrt[3]{4}}$ by $\frac{{\sqrt[3]{4}}^{2}}{{\sqrt[3]{4}}^{2}}$ .

$x = - \frac{{\sqrt[3]{4}}^{2}}{\sqrt[3]{4} {\sqrt[3]{4}}^{2}}$

Step 5.5.7.2

Raise $\sqrt[3]{4}$ to the power of $1$ .

$x = - \frac{{\sqrt[3]{4}}^{2}}{{\sqrt[3]{4}}^{1} {\sqrt[3]{4}}^{2}}$

Step 5.5.7.3

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

$x = - \frac{{\sqrt[3]{4}}^{2}}{{\sqrt[3]{4}}^{1 + 2}}$

Step 5.5.7.4

Add $1$ and $2$ .

$x = - \frac{{\sqrt[3]{4}}^{2}}{{\sqrt[3]{4}}^{3}}$

Step 5.5.7.5

Rewrite ${\sqrt[3]{4}}^{3}$ as $4$ .

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Step 5.5.7.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{4}$ as $4^{\frac{1}{3}}$ .

$x = - \frac{{\sqrt[3]{4}}^{2}}{{(4^{\frac{1}{3}})}^{3}}$

Step 5.5.7.5.2

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

$x = - \frac{{\sqrt[3]{4}}^{2}}{4^{\frac{1}{3} \cdot 3}}$

Step 5.5.7.5.3

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

$x = - \frac{{\sqrt[3]{4}}^{2}}{4^{\frac{3}{3}}}$

Step 5.5.7.5.4

Cancel the common factor of $3$ .

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Step 5.5.7.5.4.1

Cancel the common factor.

$x = - \frac{{\sqrt[3]{4}}^{2}}{4^{\frac{3}{3}}}$

Step 5.5.7.5.4.2

Rewrite the expression.

$x = - \frac{{\sqrt[3]{4}}^{2}}{4^{1}}$

Step 5.5.7.5.5

Evaluate the exponent.

$x = - \frac{{\sqrt[3]{4}}^{2}}{4}$

Step 5.5.8

Simplify the numerator.

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Step 5.5.8.1

Rewrite ${\sqrt[3]{4}}^{2}$ as $\sqrt[3]{4^{2}}$ .

$x = - \frac{\sqrt[3]{4^{2}}}{4}$

Step 5.5.8.2

Raise $4$ to the power of $2$ .

$x = - \frac{\sqrt[3]{16}}{4}$

Step 5.5.8.3

Rewrite $16$ as $2^{3} \cdot 2$ .

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Step 5.5.8.3.1

Factor $8$ out of $16$ .

$x = - \frac{\sqrt[3]{8 (2)}}{4}$

Step 5.5.8.3.2

Rewrite $8$ as $2^{3}$ .

$x = - \frac{\sqrt[3]{2^{3} \cdot 2}}{4}$

Step 5.5.8.4

Pull terms out from under the radical.

$x = - \frac{2 \sqrt[3]{2}}{4}$

Step 5.5.9

Cancel the common factor of $2$ and $4$ .

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Step 5.5.9.1

Factor $2$ out of $2 \sqrt[3]{2}$ .

$x = - \frac{2 (\sqrt[3]{2})}{4}$

Step 5.5.9.2

Cancel the common factors.

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Step 5.5.9.2.1

Factor $2$ out of $4$ .

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

Step 5.5.9.2.2

Cancel the common factor.

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

Step 5.5.9.2.3

Rewrite the expression.

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

Step 6

Find the values where the derivative is undefined.

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Step 6.1

The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.

Step 7

Critical points to evaluate.

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

Step 8

Evaluate the second derivative at $x = - \frac{\sqrt[3]{2}}{2}$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$12 {(- \frac{\sqrt[3]{2}}{2})}^{2}$

Step 9

Evaluate the second derivative.

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Step 9.1

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 9.1.1

Apply the product rule to $- \frac{\sqrt[3]{2}}{2}$ .

$12 ({(- 1)}^{2} {(\frac{\sqrt[3]{2}}{2})}^{2})$

Step 9.1.2

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

$12 ({(- 1)}^{2} \frac{{\sqrt[3]{2}}^{2}}{2^{2}})$

Step 9.2

Simplify the expression.

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Step 9.2.1

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

$12 (1 \frac{{\sqrt[3]{2}}^{2}}{2^{2}})$

Step 9.2.2

Multiply $\frac{{\sqrt[3]{2}}^{2}}{2^{2}}$ by $1$ .

$12 \frac{{\sqrt[3]{2}}^{2}}{2^{2}}$

Step 9.3

Simplify the numerator.

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Step 9.3.1

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

$12 \frac{\sqrt[3]{2^{2}}}{2^{2}}$

Step 9.3.2

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

$12 \frac{\sqrt[3]{4}}{2^{2}}$

Step 9.4

Reduce the expression by cancelling the common factors.

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Step 9.4.1

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

$12 \frac{\sqrt[3]{4}}{4}$

Step 9.4.2

Cancel the common factor of $4$ .

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Step 9.4.2.1

Factor $4$ out of $12$ .

$4 (3) \frac{\sqrt[3]{4}}{4}$

Step 9.4.2.2

Cancel the common factor.

$4 \cdot 3 \frac{\sqrt[3]{4}}{4}$

Step 9.4.2.3

Rewrite the expression.

$3 \sqrt[3]{4}$

Step 10

$x = - \frac{\sqrt[3]{2}}{2}$ is a local minimum because the value of the second derivative is positive. This is referred to as the second derivative test.

$x = - \frac{\sqrt[3]{2}}{2}$ is a local minimum

Step 11

Find the y-value when $x = - \frac{\sqrt[3]{2}}{2}$ .

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Step 11.1

Replace the variable $x$ with $- \frac{\sqrt[3]{2}}{2}$ in the expression.

$f (- \frac{\sqrt[3]{2}}{2}) = {(- \frac{\sqrt[3]{2}}{2})}^{4} - \frac{\sqrt[3]{2}}{2} + 2$

Step 11.2

Simplify the result.

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Step 11.2.1

Remove parentheses.

$f (- \frac{\sqrt[3]{2}}{2}) = {(- \frac{\sqrt[3]{2}}{2})}^{4} - \frac{\sqrt[3]{2}}{2} + 2$

Step 11.2.2

Simplify each term.

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Step 11.2.2.1

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 11.2.2.1.1

Apply the product rule to $- \frac{\sqrt[3]{2}}{2}$ .

$f (- \frac{\sqrt[3]{2}}{2}) = {(- 1)}^{4} {(\frac{\sqrt[3]{2}}{2})}^{4} - \frac{\sqrt[3]{2}}{2} + 2$

Step 11.2.2.1.2

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

$f (- \frac{\sqrt[3]{2}}{2}) = {(- 1)}^{4} (\frac{{\sqrt[3]{2}}^{4}}{2^{4}}) - \frac{\sqrt[3]{2}}{2} + 2$

Step 11.2.2.2

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

$f (- \frac{\sqrt[3]{2}}{2}) = 1 (\frac{{\sqrt[3]{2}}^{4}}{2^{4}}) - \frac{\sqrt[3]{2}}{2} + 2$

Step 11.2.2.3

Multiply $\frac{{\sqrt[3]{2}}^{4}}{2^{4}}$ by $1$ .

$f (- \frac{\sqrt[3]{2}}{2}) = \frac{{\sqrt[3]{2}}^{4}}{2^{4}} - \frac{\sqrt[3]{2}}{2} + 2$

Step 11.2.2.4

Simplify the numerator.

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Step 11.2.2.4.1

Rewrite ${\sqrt[3]{2}}^{4}$ as $\sqrt[3]{2^{4}}$ .

$f (- \frac{\sqrt[3]{2}}{2}) = \frac{\sqrt[3]{2^{4}}}{2^{4}} - \frac{\sqrt[3]{2}}{2} + 2$

Step 11.2.2.4.2

Raise $2$ to the power of $4$ .

$f (- \frac{\sqrt[3]{2}}{2}) = \frac{\sqrt[3]{16}}{2^{4}} - \frac{\sqrt[3]{2}}{2} + 2$

Step 11.2.2.4.3

Rewrite $16$ as $2^{3} \cdot 2$ .

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Step 11.2.2.4.3.1

Factor $8$ out of $16$ .

$f (- \frac{\sqrt[3]{2}}{2}) = \frac{\sqrt[3]{8 (2)}}{2^{4}} - \frac{\sqrt[3]{2}}{2} + 2$

Step 11.2.2.4.3.2

Rewrite $8$ as $2^{3}$ .

$f (- \frac{\sqrt[3]{2}}{2}) = \frac{\sqrt[3]{2^{3} \cdot 2}}{2^{4}} - \frac{\sqrt[3]{2}}{2} + 2$

Step 11.2.2.4.4

Pull terms out from under the radical.

$f (- \frac{\sqrt[3]{2}}{2}) = \frac{2 \sqrt[3]{2}}{2^{4}} - \frac{\sqrt[3]{2}}{2} + 2$

Step 11.2.2.5

Raise $2$ to the power of $4$ .

$f (- \frac{\sqrt[3]{2}}{2}) = \frac{2 \sqrt[3]{2}}{16} - \frac{\sqrt[3]{2}}{2} + 2$

Step 11.2.2.6

Cancel the common factor of $2$ and $16$ .

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Step 11.2.2.6.1

Factor $2$ out of $2 \sqrt[3]{2}$ .

$f (- \frac{\sqrt[3]{2}}{2}) = \frac{2 (\sqrt[3]{2})}{16} - \frac{\sqrt[3]{2}}{2} + 2$

Step 11.2.2.6.2

Cancel the common factors.

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Step 11.2.2.6.2.1

Factor $2$ out of $16$ .

$f (- \frac{\sqrt[3]{2}}{2}) = \frac{2 \sqrt[3]{2}}{2 \cdot 8} - \frac{\sqrt[3]{2}}{2} + 2$

Step 11.2.2.6.2.2

Cancel the common factor.

$f (- \frac{\sqrt[3]{2}}{2}) = \frac{2 \sqrt[3]{2}}{2 \cdot 8} - \frac{\sqrt[3]{2}}{2} + 2$

Step 11.2.2.6.2.3

Rewrite the expression.

$f (- \frac{\sqrt[3]{2}}{2}) = \frac{\sqrt[3]{2}}{8} - \frac{\sqrt[3]{2}}{2} + 2$

Step 11.2.3

To write $- \frac{\sqrt[3]{2}}{2}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$f (- \frac{\sqrt[3]{2}}{2}) = \frac{\sqrt[3]{2}}{8} - \frac{\sqrt[3]{2}}{2} \cdot \frac{4}{4} + 2$

Step 11.2.4

Write each expression with a common denominator of $8$ , by multiplying each by an appropriate factor of $1$ .

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Step 11.2.4.1

Multiply $\frac{\sqrt[3]{2}}{2}$ by $\frac{4}{4}$ .

$f (- \frac{\sqrt[3]{2}}{2}) = \frac{\sqrt[3]{2}}{8} - \frac{\sqrt[3]{2} \cdot 4}{2 \cdot 4} + 2$

Step 11.2.4.2

Multiply $2$ by $4$ .

$f (- \frac{\sqrt[3]{2}}{2}) = \frac{\sqrt[3]{2}}{8} - \frac{\sqrt[3]{2} \cdot 4}{8} + 2$

Step 11.2.5

Combine the numerators over the common denominator.

$f (- \frac{\sqrt[3]{2}}{2}) = \frac{\sqrt[3]{2} - \sqrt[3]{2} \cdot 4}{8} + 2$

Step 11.2.6

Simplify each term.

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Step 11.2.6.1

Simplify the numerator.

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Step 11.2.6.1.1

Multiply $4$ by $- 1$ .

$f (- \frac{\sqrt[3]{2}}{2}) = \frac{\sqrt[3]{2} - 4 \sqrt[3]{2}}{8} + 2$

Step 11.2.6.1.2

Subtract $4 \sqrt[3]{2}$ from $\sqrt[3]{2}$ .

$f (- \frac{\sqrt[3]{2}}{2}) = \frac{- 3 \sqrt[3]{2}}{8} + 2$

Step 11.2.6.2

Move the negative in front of the fraction.

$f (- \frac{\sqrt[3]{2}}{2}) = - \frac{3 \sqrt[3]{2}}{8} + 2$

Step 11.2.7

The final answer is $- \frac{3 \sqrt[3]{2}}{8} + 2$ .

$y = - \frac{3 \sqrt[3]{2}}{8} + 2$

Step 12

These are the local extrema for $f (x) = x^{4} + x + 2$ .

$(- \frac{\sqrt[3]{2}}{2}, - \frac{3 \sqrt[3]{2}}{8} + 2)$ is a local minima

Step 13