Calculus Examples

$f (x) = 5 x^{4} - 16 x$

Step 1

Find the first 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 $5 x^{4} - 16 x$ with respect to $x$ is $\frac{d}{d x} [5 x^{4}] + \frac{d}{d x} [- 16 x]$ .

$\frac{d}{d x} [5 x^{4}] + \frac{d}{d x} [- 16 x]$

Step 1.1.2

Evaluate $\frac{d}{d x} [5 x^{4}]$ .

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Step 1.1.2.1

Since $5$ is constant with respect to $x$ , the derivative of $5 x^{4}$ with respect to $x$ is $5 \frac{d}{d x} [x^{4}]$ .

$5 \frac{d}{d x} [x^{4}] + \frac{d}{d x} [- 16 x]$

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 $n = 4$ .

$5 (4 x^{3}) + \frac{d}{d x} [- 16 x]$

Step 1.1.2.3

Multiply $4$ by $5$ .

$20 x^{3} + \frac{d}{d x} [- 16 x]$

Step 1.1.3

Evaluate $\frac{d}{d x} [- 16 x]$ .

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Step 1.1.3.1

Since $- 16$ is constant with respect to $x$ , the derivative of $- 16 x$ with respect to $x$ is $- 16 \frac{d}{d x} [x]$ .

$20 x^{3} - 16 \frac{d}{d x} [x]$

Step 1.1.3.2

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

$20 x^{3} - 16 \cdot 1$

Step 1.1.3.3

Multiply $- 16$ by $1$ .

$f' (x) = 20 x^{3} - 16$

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $20 x^{3} - 16$ .

$20 x^{3} - 16$

Step 2

Set the first derivative equal to $0$ then solve the equation $20 x^{3} - 16 = 0$ .

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Step 2.1

Set the first derivative equal to $0$ .

$20 x^{3} - 16 = 0$

Step 2.2

Add $16$ to both sides of the equation.

$20 x^{3} = 16$

Step 2.3

Subtract $16$ from both sides of the equation.

$20 x^{3} - 16 = 0$

Step 2.4

Factor $4$ out of $20 x^{3} - 16$ .

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Step 2.4.1

Factor $4$ out of $20 x^{3}$ .

$4 (5 x^{3}) - 16 = 0$

Step 2.4.2

Factor $4$ out of $- 16$ .

$4 (5 x^{3}) + 4 (- 4) = 0$

Step 2.4.3

Factor $4$ out of $4 (5 x^{3}) + 4 (- 4)$ .

$4 (5 x^{3} - 4) = 0$

Step 2.5

Divide each term in $4 (5 x^{3} - 4) = 0$ by $4$ and simplify.

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Step 2.5.1

Divide each term in $4 (5 x^{3} - 4) = 0$ by $4$ .

$\frac{4 (5 x^{3} - 4)}{4} = \frac{0}{4}$

Step 2.5.2

Simplify the left side.

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Step 2.5.2.1

Cancel the common factor of $4$ .

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Step 2.5.2.1.1

Cancel the common factor.

$\frac{4 (5 x^{3} - 4)}{4} = \frac{0}{4}$

Step 2.5.2.1.2

Divide $5 x^{3} - 4$ by $1$ .

$5 x^{3} - 4 = \frac{0}{4}$

Step 2.5.3

Simplify the right side.

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Step 2.5.3.1

Divide $0$ by $4$ .

$5 x^{3} - 4 = 0$

Step 2.6

Add $4$ to both sides of the equation.

$5 x^{3} = 4$

Step 2.7

Divide each term in $5 x^{3} = 4$ by $5$ and simplify.

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Step 2.7.1

Divide each term in $5 x^{3} = 4$ by $5$ .

$\frac{5 x^{3}}{5} = \frac{4}{5}$

Step 2.7.2

Simplify the left side.

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Step 2.7.2.1

Cancel the common factor of $5$ .

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Step 2.7.2.1.1

Cancel the common factor.

$\frac{5 x^{3}}{5} = \frac{4}{5}$

Step 2.7.2.1.2

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

$x^{3} = \frac{4}{5}$

Step 2.8

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

$x = \sqrt[3]{\frac{4}{5}}$

Step 2.9

Simplify $\sqrt[3]{\frac{4}{5}}$ .

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Step 2.9.1

Rewrite $\sqrt[3]{\frac{4}{5}}$ as $\frac{\sqrt[3]{4}}{\sqrt[3]{5}}$ .

$x = \frac{\sqrt[3]{4}}{\sqrt[3]{5}}$

Step 2.9.2

Multiply $\frac{\sqrt[3]{4}}{\sqrt[3]{5}}$ by $\frac{{\sqrt[3]{5}}^{2}}{{\sqrt[3]{5}}^{2}}$ .

$x = \frac{\sqrt[3]{4}}{\sqrt[3]{5}} \cdot \frac{{\sqrt[3]{5}}^{2}}{{\sqrt[3]{5}}^{2}}$

Step 2.9.3

Combine and simplify the denominator.

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Step 2.9.3.1

Multiply $\frac{\sqrt[3]{4}}{\sqrt[3]{5}}$ by $\frac{{\sqrt[3]{5}}^{2}}{{\sqrt[3]{5}}^{2}}$ .

$x = \frac{\sqrt[3]{4} {\sqrt[3]{5}}^{2}}{\sqrt[3]{5} {\sqrt[3]{5}}^{2}}$

Step 2.9.3.2

Raise $\sqrt[3]{5}$ to the power of $1$ .

$x = \frac{\sqrt[3]{4} {\sqrt[3]{5}}^{2}}{{\sqrt[3]{5}}^{1} {\sqrt[3]{5}}^{2}}$

Step 2.9.3.3

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

$x = \frac{\sqrt[3]{4} {\sqrt[3]{5}}^{2}}{{\sqrt[3]{5}}^{1 + 2}}$

Step 2.9.3.4

Add $1$ and $2$ .

$x = \frac{\sqrt[3]{4} {\sqrt[3]{5}}^{2}}{{\sqrt[3]{5}}^{3}}$

Step 2.9.3.5

Rewrite ${\sqrt[3]{5}}^{3}$ as $5$ .

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Step 2.9.3.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{5}$ as $5^{\frac{1}{3}}$ .

$x = \frac{\sqrt[3]{4} {\sqrt[3]{5}}^{2}}{{(5^{\frac{1}{3}})}^{3}}$

Step 2.9.3.5.2

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

$x = \frac{\sqrt[3]{4} {\sqrt[3]{5}}^{2}}{5^{\frac{1}{3} \cdot 3}}$

Step 2.9.3.5.3

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

$x = \frac{\sqrt[3]{4} {\sqrt[3]{5}}^{2}}{5^{\frac{3}{3}}}$

Step 2.9.3.5.4

Cancel the common factor of $3$ .

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Step 2.9.3.5.4.1

Cancel the common factor.

$x = \frac{\sqrt[3]{4} {\sqrt[3]{5}}^{2}}{5^{\frac{3}{3}}}$

Step 2.9.3.5.4.2

Rewrite the expression.

$x = \frac{\sqrt[3]{4} {\sqrt[3]{5}}^{2}}{5^{1}}$

Step 2.9.3.5.5

Evaluate the exponent.

$x = \frac{\sqrt[3]{4} {\sqrt[3]{5}}^{2}}{5}$

Step 2.9.4

Simplify the numerator.

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Step 2.9.4.1

Rewrite ${\sqrt[3]{5}}^{2}$ as $\sqrt[3]{5^{2}}$ .

$x = \frac{\sqrt[3]{4} \sqrt[3]{5^{2}}}{5}$

Step 2.9.4.2

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

$x = \frac{\sqrt[3]{4} \sqrt[3]{25}}{5}$

Step 2.9.5

Simplify the numerator.

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Step 2.9.5.1

Combine using the product rule for radicals.

$x = \frac{\sqrt[3]{4 \cdot 25}}{5}$

Step 2.9.5.2

Multiply $4$ by $25$ .

$x = \frac{\sqrt[3]{100}}{5}$

Step 3

Find the values where the derivative is undefined.

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Step 3.1

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 4

Evaluate $5 x^{4} - 16 x$ at each $x$ value where the derivative is $0$ or undefined.

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Step 4.1

Evaluate at $x = \frac{\sqrt[3]{100}}{5}$ .

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Step 4.1.1

Substitute $\frac{\sqrt[3]{100}}{5}$ for $x$ .

$5 {(\frac{\sqrt[3]{100}}{5})}^{4} - 16 \frac{\sqrt[3]{100}}{5}$

Step 4.1.2

Simplify.

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Step 4.1.2.1

Simplify each term.

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Step 4.1.2.1.1

Apply the product rule to $\frac{\sqrt[3]{100}}{5}$ .

$5 \frac{{\sqrt[3]{100}}^{4}}{5^{4}} - 16 \frac{\sqrt[3]{100}}{5}$

Step 4.1.2.1.2

Simplify the numerator.

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Step 4.1.2.1.2.1

Rewrite ${\sqrt[3]{100}}^{4}$ as $\sqrt[3]{100^{4}}$ .

$5 \frac{\sqrt[3]{100^{4}}}{5^{4}} - 16 \frac{\sqrt[3]{100}}{5}$

Step 4.1.2.1.2.2

Raise $100$ to the power of $4$ .

$5 \frac{\sqrt[3]{100000000}}{5^{4}} - 16 \frac{\sqrt[3]{100}}{5}$

Step 4.1.2.1.2.3

Rewrite $100000000$ as $100^{3} \cdot 100$ .

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Step 4.1.2.1.2.3.1

Factor $1000000$ out of $100000000$ .

$5 \frac{\sqrt[3]{1000000 (100)}}{5^{4}} - 16 \frac{\sqrt[3]{100}}{5}$

Step 4.1.2.1.2.3.2

Rewrite $1000000$ as $100^{3}$ .

$5 \frac{\sqrt[3]{100^{3} \cdot 100}}{5^{4}} - 16 \frac{\sqrt[3]{100}}{5}$

Step 4.1.2.1.2.4

Pull terms out from under the radical.

$5 \frac{100 \sqrt[3]{100}}{5^{4}} - 16 \frac{\sqrt[3]{100}}{5}$

Step 4.1.2.1.3

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

$5 \frac{100 \sqrt[3]{100}}{625} - 16 \frac{\sqrt[3]{100}}{5}$

Step 4.1.2.1.4

Cancel the common factor of $5$ .

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Step 4.1.2.1.4.1

Factor $5$ out of $625$ .

$5 \frac{100 \sqrt[3]{100}}{5 (125)} - 16 \frac{\sqrt[3]{100}}{5}$

Step 4.1.2.1.4.2

Cancel the common factor.

$5 \frac{100 \sqrt[3]{100}}{5 \cdot 125} - 16 \frac{\sqrt[3]{100}}{5}$

Step 4.1.2.1.4.3

Rewrite the expression.

$\frac{100 \sqrt[3]{100}}{125} - 16 \frac{\sqrt[3]{100}}{5}$

Step 4.1.2.1.5

Cancel the common factor of $100$ and $125$ .

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Step 4.1.2.1.5.1

Factor $25$ out of $100 \sqrt[3]{100}$ .

$\frac{25 (4 \sqrt[3]{100})}{125} - 16 \frac{\sqrt[3]{100}}{5}$

Step 4.1.2.1.5.2

Cancel the common factors.

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Step 4.1.2.1.5.2.1

Factor $25$ out of $125$ .

$\frac{25 (4 \sqrt[3]{100})}{25 (5)} - 16 \frac{\sqrt[3]{100}}{5}$

Step 4.1.2.1.5.2.2

Cancel the common factor.

$\frac{25 (4 \sqrt[3]{100})}{25 \cdot 5} - 16 \frac{\sqrt[3]{100}}{5}$

Step 4.1.2.1.5.2.3

Rewrite the expression.

$\frac{4 \sqrt[3]{100}}{5} - 16 \frac{\sqrt[3]{100}}{5}$

Step 4.1.2.1.6

Combine $- 16$ and $\frac{\sqrt[3]{100}}{5}$ .

$\frac{4 \sqrt[3]{100}}{5} + \frac{- 16 \sqrt[3]{100}}{5}$

Step 4.1.2.1.7

Move the negative in front of the fraction.

$\frac{4 \sqrt[3]{100}}{5} - \frac{16 \sqrt[3]{100}}{5}$

Step 4.1.2.2

Simplify terms.

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Step 4.1.2.2.1

Combine the numerators over the common denominator.

$\frac{4 \sqrt[3]{100} - 16 \sqrt[3]{100}}{5}$

Step 4.1.2.2.2

Subtract $16 \sqrt[3]{100}$ from $4 \sqrt[3]{100}$ .

$\frac{- 12 \sqrt[3]{100}}{5}$

Step 4.1.2.2.3

Move the negative in front of the fraction.

$- \frac{12 \sqrt[3]{100}}{5}$

Step 4.2

List all of the points.

$(\frac{\sqrt[3]{100}}{5}, - \frac{12 \sqrt[3]{100}}{5})$

Step 5