Calculus Examples

$f (x) = x^{2} + \frac{240}{x}$ ; $(0, \infty)$

Step 1

Find the critical points.

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Step 1.1

Find the first derivative.

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Step 1.1.1

Find the first derivative.

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Step 1.1.1.1

Differentiate.

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Step 1.1.1.1.1

By the Sum Rule, the derivative of $x^{2} + \frac{240}{x}$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [\frac{240}{x}]$ .

$\frac{d}{d x} [x^{2}] + \frac{d}{d x} [\frac{240}{x}]$

Step 1.1.1.1.2

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

$2 x + \frac{d}{d x} [\frac{240}{x}]$

Step 1.1.1.2

Evaluate $\frac{d}{d x} [\frac{240}{x}]$ .

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Step 1.1.1.2.1

Since $240$ is constant with respect to $x$ , the derivative of $\frac{240}{x}$ with respect to $x$ is $240 \frac{d}{d x} [\frac{1}{x}]$ .

$2 x + 240 \frac{d}{d x} [\frac{1}{x}]$

Step 1.1.1.2.2

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

$2 x + 240 \frac{d}{d x} [x^{- 1}]$

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 $n = - 1$ .

$2 x + 240 (- x^{- 2})$

Step 1.1.1.2.4

Multiply $- 1$ by $240$ .

$2 x - 240 x^{- 2}$

Step 1.1.1.3

Simplify.

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Step 1.1.1.3.1

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

$2 x - 240 \frac{1}{x^{2}}$

Step 1.1.1.3.2

Combine terms.

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Step 1.1.1.3.2.1

Combine $- 240$ and $\frac{1}{x^{2}}$ .

$2 x + \frac{- 240}{x^{2}}$

Step 1.1.1.3.2.2

Move the negative in front of the fraction.

$f' (x) = 2 x - \frac{240}{x^{2}}$

Step 1.1.2

The first derivative of $f (x)$ with respect to $x$ is $2 x - \frac{240}{x^{2}}$ .

$2 x - \frac{240}{x^{2}}$

Step 1.2

Set the first derivative equal to $0$ then solve the equation $2 x - \frac{240}{x^{2}} = 0$ .

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Step 1.2.1

Set the first derivative equal to $0$ .

$2 x - \frac{240}{x^{2}} = 0$

Step 1.2.2

Find the LCD of the terms in the equation.

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Step 1.2.2.1

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

$1, x^{2}, 1$

Step 1.2.2.2

The LCM of one and any expression is the expression.

$x^{2}$

Step 1.2.3

Multiply each term in $2 x - \frac{240}{x^{2}} = 0$ by $x^{2}$ to eliminate the fractions.

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Step 1.2.3.1

Multiply each term in $2 x - \frac{240}{x^{2}} = 0$ by $x^{2}$ .

$2 x \cdot x^{2} - \frac{240}{x^{2}} x^{2} = 0 x^{2}$

Step 1.2.3.2

Simplify the left side.

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Step 1.2.3.2.1

Simplify each term.

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Step 1.2.3.2.1.1

Multiply $x$ by $x^{2}$ by adding the exponents.

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Step 1.2.3.2.1.1.1

Move $x^{2}$ .

$2 (x^{2} x) - \frac{240}{x^{2}} x^{2} = 0 x^{2}$

Step 1.2.3.2.1.1.2

Multiply $x^{2}$ by $x$ .

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Step 1.2.3.2.1.1.2.1

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

$2 (x^{2} x^{1}) - \frac{240}{x^{2}} x^{2} = 0 x^{2}$

Step 1.2.3.2.1.1.2.2

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

$2 x^{2 + 1} - \frac{240}{x^{2}} x^{2} = 0 x^{2}$

Step 1.2.3.2.1.1.3

Add $2$ and $1$ .

$2 x^{3} - \frac{240}{x^{2}} x^{2} = 0 x^{2}$

Step 1.2.3.2.1.2

Cancel the common factor of $x^{2}$ .

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Step 1.2.3.2.1.2.1

Move the leading negative in $- \frac{240}{x^{2}}$ into the numerator.

$2 x^{3} + \frac{- 240}{x^{2}} x^{2} = 0 x^{2}$

Step 1.2.3.2.1.2.2

Cancel the common factor.

$2 x^{3} + \frac{- 240}{x^{2}} x^{2} = 0 x^{2}$

Step 1.2.3.2.1.2.3

Rewrite the expression.

$2 x^{3} - 240 = 0 x^{2}$

Step 1.2.3.3

Simplify the right side.

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Step 1.2.3.3.1

Multiply $0$ by $x^{2}$ .

$2 x^{3} - 240 = 0$

Step 1.2.4

Solve the equation.

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Step 1.2.4.1

Add $240$ to both sides of the equation.

$2 x^{3} = 240$

Step 1.2.4.2

Subtract $240$ from both sides of the equation.

$2 x^{3} - 240 = 0$

Step 1.2.4.3

Factor $2$ out of $2 x^{3} - 240$ .

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Step 1.2.4.3.1

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

$2 (x^{3}) - 240 = 0$

Step 1.2.4.3.2

Factor $2$ out of $- 240$ .

$2 x^{3} + 2 \cdot - 120 = 0$

Step 1.2.4.3.3

Factor $2$ out of $2 x^{3} + 2 \cdot - 120$ .

$2 (x^{3} - 120) = 0$

Step 1.2.4.4

Divide each term in $2 (x^{3} - 120) = 0$ by $2$ and simplify.

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Step 1.2.4.4.1

Divide each term in $2 (x^{3} - 120) = 0$ by $2$ .

$\frac{2 (x^{3} - 120)}{2} = \frac{0}{2}$

Step 1.2.4.4.2

Simplify the left side.

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Step 1.2.4.4.2.1

Cancel the common factor of $2$ .

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Step 1.2.4.4.2.1.1

Cancel the common factor.

$\frac{2 (x^{3} - 120)}{2} = \frac{0}{2}$

Step 1.2.4.4.2.1.2

Divide $x^{3} - 120$ by $1$ .

$x^{3} - 120 = \frac{0}{2}$

Step 1.2.4.4.3

Simplify the right side.

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Step 1.2.4.4.3.1

Divide $0$ by $2$ .

$x^{3} - 120 = 0$

Step 1.2.4.5

Add $120$ to both sides of the equation.

$x^{3} = 120$

Step 1.2.4.6

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

$x = \sqrt[3]{120}$

Step 1.2.4.7

Simplify $\sqrt[3]{120}$ .

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Step 1.2.4.7.1

Rewrite $120$ as $2^{3} \cdot 15$ .

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Step 1.2.4.7.1.1

Factor $8$ out of $120$ .

$x = \sqrt[3]{8 (15)}$

Step 1.2.4.7.1.2

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

$x = \sqrt[3]{2^{3} \cdot 15}$

Step 1.2.4.7.2

Pull terms out from under the radical.

$x = 2 \sqrt[3]{15}$

Step 1.3

Find the values where the derivative is undefined.

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Step 1.3.1

Set the denominator in $\frac{240}{x^{2}}$ equal to $0$ to find where the expression is undefined.

$x^{2} = 0$

Step 1.3.2

Solve for $x$ .

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Step 1.3.2.1

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

$x = \pm \sqrt{0}$

Step 1.3.2.2

Simplify $\pm \sqrt{0}$ .

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Step 1.3.2.2.1

Rewrite $0$ as $0^{2}$ .

$x = \pm \sqrt{0^{2}}$

Step 1.3.2.2.2

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

$x = \pm 0$

Step 1.3.2.2.3

Plus or minus $0$ is $0$ .

$x = 0$

Step 1.4

Evaluate $x^{2} + \frac{240}{x}$ at each $x$ value where the derivative is $0$ or undefined.

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Step 1.4.1

Evaluate at $x = 2 \sqrt[3]{15}$ .

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Step 1.4.1.1

Substitute $2 \sqrt[3]{15}$ for $x$ .

${(2 \sqrt[3]{15})}^{2} + \frac{240}{2 \sqrt[3]{15}}$

Step 1.4.1.2

Simplify.

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Step 1.4.1.2.1

Simplify each term.

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Step 1.4.1.2.1.1

Apply the product rule to $2 \sqrt[3]{15}$ .

$2^{2} {\sqrt[3]{15}}^{2} + \frac{240}{2 \sqrt[3]{15}}$

Step 1.4.1.2.1.2

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

$4 {\sqrt[3]{15}}^{2} + \frac{240}{2 \sqrt[3]{15}}$

Step 1.4.1.2.1.3

Rewrite ${\sqrt[3]{15}}^{2}$ as $\sqrt[3]{15^{2}}$ .

$4 \sqrt[3]{15^{2}} + \frac{240}{2 \sqrt[3]{15}}$

Step 1.4.1.2.1.4

Raise $15$ to the power of $2$ .

$4 \sqrt[3]{225} + \frac{240}{2 \sqrt[3]{15}}$

Step 1.4.1.2.1.5

Cancel the common factor of $240$ and $2$ .

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Step 1.4.1.2.1.5.1

Factor $2$ out of $240$ .

$4 \sqrt[3]{225} + \frac{2 \cdot 120}{2 \sqrt[3]{15}}$

Step 1.4.1.2.1.5.2

Cancel the common factors.

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Step 1.4.1.2.1.5.2.1

Factor $2$ out of $2 \sqrt[3]{15}$ .

$4 \sqrt[3]{225} + \frac{2 \cdot 120}{2 (\sqrt[3]{15})}$

Step 1.4.1.2.1.5.2.2

Cancel the common factor.

$4 \sqrt[3]{225} + \frac{2 \cdot 120}{2 \sqrt[3]{15}}$

Step 1.4.1.2.1.5.2.3

Rewrite the expression.

$4 \sqrt[3]{225} + \frac{120}{\sqrt[3]{15}}$

Step 1.4.1.2.1.6

Multiply $\frac{120}{\sqrt[3]{15}}$ by $\frac{{\sqrt[3]{15}}^{2}}{{\sqrt[3]{15}}^{2}}$ .

$4 \sqrt[3]{225} + \frac{120}{\sqrt[3]{15}} \cdot \frac{{\sqrt[3]{15}}^{2}}{{\sqrt[3]{15}}^{2}}$

Step 1.4.1.2.1.7

Combine and simplify the denominator.

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Step 1.4.1.2.1.7.1

Multiply $\frac{120}{\sqrt[3]{15}}$ by $\frac{{\sqrt[3]{15}}^{2}}{{\sqrt[3]{15}}^{2}}$ .

$4 \sqrt[3]{225} + \frac{120 {\sqrt[3]{15}}^{2}}{\sqrt[3]{15} {\sqrt[3]{15}}^{2}}$

Step 1.4.1.2.1.7.2

Raise $\sqrt[3]{15}$ to the power of $1$ .

$4 \sqrt[3]{225} + \frac{120 {\sqrt[3]{15}}^{2}}{{\sqrt[3]{15}}^{1} {\sqrt[3]{15}}^{2}}$

Step 1.4.1.2.1.7.3

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

$4 \sqrt[3]{225} + \frac{120 {\sqrt[3]{15}}^{2}}{{\sqrt[3]{15}}^{1 + 2}}$

Step 1.4.1.2.1.7.4

Add $1$ and $2$ .

$4 \sqrt[3]{225} + \frac{120 {\sqrt[3]{15}}^{2}}{{\sqrt[3]{15}}^{3}}$

Step 1.4.1.2.1.7.5

Rewrite ${\sqrt[3]{15}}^{3}$ as $15$ .

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Step 1.4.1.2.1.7.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{15}$ as $15^{\frac{1}{3}}$ .

$4 \sqrt[3]{225} + \frac{120 {\sqrt[3]{15}}^{2}}{{(15^{\frac{1}{3}})}^{3}}$

Step 1.4.1.2.1.7.5.2

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

$4 \sqrt[3]{225} + \frac{120 {\sqrt[3]{15}}^{2}}{15^{\frac{1}{3} \cdot 3}}$

Step 1.4.1.2.1.7.5.3

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

$4 \sqrt[3]{225} + \frac{120 {\sqrt[3]{15}}^{2}}{15^{\frac{3}{3}}}$

Step 1.4.1.2.1.7.5.4

Cancel the common factor of $3$ .

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Step 1.4.1.2.1.7.5.4.1

Cancel the common factor.

$4 \sqrt[3]{225} + \frac{120 {\sqrt[3]{15}}^{2}}{15^{\frac{3}{3}}}$

Step 1.4.1.2.1.7.5.4.2

Rewrite the expression.

$4 \sqrt[3]{225} + \frac{120 {\sqrt[3]{15}}^{2}}{15^{1}}$

Step 1.4.1.2.1.7.5.5

Evaluate the exponent.

$4 \sqrt[3]{225} + \frac{120 {\sqrt[3]{15}}^{2}}{15}$

Step 1.4.1.2.1.8

Cancel the common factor of $120$ and $15$ .

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Step 1.4.1.2.1.8.1

Factor $15$ out of $120 {\sqrt[3]{15}}^{2}$ .

$4 \sqrt[3]{225} + \frac{15 (8 {\sqrt[3]{15}}^{2})}{15}$

Step 1.4.1.2.1.8.2

Cancel the common factors.

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Step 1.4.1.2.1.8.2.1

Factor $15$ out of $15$ .

$4 \sqrt[3]{225} + \frac{15 (8 {\sqrt[3]{15}}^{2})}{15 (1)}$

Step 1.4.1.2.1.8.2.2

Cancel the common factor.

$4 \sqrt[3]{225} + \frac{15 (8 {\sqrt[3]{15}}^{2})}{15 \cdot 1}$

Step 1.4.1.2.1.8.2.3

Rewrite the expression.

$4 \sqrt[3]{225} + \frac{8 {\sqrt[3]{15}}^{2}}{1}$

Step 1.4.1.2.1.8.2.4

Divide $8 {\sqrt[3]{15}}^{2}$ by $1$ .

$4 \sqrt[3]{225} + 8 {\sqrt[3]{15}}^{2}$

Step 1.4.1.2.1.9

Rewrite ${\sqrt[3]{15}}^{2}$ as $\sqrt[3]{15^{2}}$ .

$4 \sqrt[3]{225} + 8 \sqrt[3]{15^{2}}$

Step 1.4.1.2.1.10

Raise $15$ to the power of $2$ .

$4 \sqrt[3]{225} + 8 \sqrt[3]{225}$

Step 1.4.1.2.2

Add $4 \sqrt[3]{225}$ and $8 \sqrt[3]{225}$ .

$12 \sqrt[3]{225}$

Step 1.4.2

Evaluate at $x = 0$ .

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Step 1.4.2.1

Substitute $0$ for $x$ .

${(0)}^{2} + \frac{240}{0}$

Step 1.4.2.2

The expression contains a division by $0$ . The expression is undefined.

Undefined

Step 1.4.3

List all of the points.

$(2 \sqrt[3]{15}, 12 \sqrt[3]{225})$

Step 2

Use the first derivative test to determine which points can be maxima or minima.

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Step 2.1

Split $(- \infty, \infty)$ into separate intervals around the $x$ values that make the first derivative $0$ or undefined.

$(- \infty, 2 \sqrt[3]{15}) \cup (2 \sqrt[3]{15}, \infty)$

Step 2.2

Substitute any number, such as $- 2$ , from the interval $(- \infty, 2 \sqrt[3]{15})$ in the first derivative $2 x - \frac{240}{x^{2}}$ to check if the result is negative or positive.

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Step 2.2.1

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

$f' (- 2) = 2 (- 2) - \frac{240}{{(- 2)}^{2}}$

Step 2.2.2

Simplify the result.

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Step 2.2.2.1

Simplify each term.

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Step 2.2.2.1.1

Multiply $2$ by $- 2$ .

$f' (- 2) = - 4 - \frac{240}{{(- 2)}^{2}}$

Step 2.2.2.1.2

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

$f' (- 2) = - 4 - \frac{240}{4}$

Step 2.2.2.1.3

Divide $240$ by $4$ .

$f' (- 2) = - 4 - 1 \cdot 60$

Step 2.2.2.1.4

Multiply $- 1$ by $60$ .

$f' (- 2) = - 4 - 60$

Step 2.2.2.2

Subtract $60$ from $- 4$ .

$f' (- 2) = - 64$

Step 2.2.2.3

The final answer is $- 64$ .

$- 64$

Step 2.3

Substitute any number, such as $7$ , from the interval $(2 \sqrt[3]{15}, \infty)$ in the first derivative $2 x - \frac{240}{x^{2}}$ to check if the result is negative or positive.

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Step 2.3.1

Replace the variable $x$ with $7$ in the expression.

$f' (7) = 2 (7) - \frac{240}{{(7)}^{2}}$

Step 2.3.2

Simplify the result.

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Step 2.3.2.1

Simplify each term.

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Step 2.3.2.1.1

Multiply $2$ by $7$ .

$f' (7) = 14 - \frac{240}{{(7)}^{2}}$

Step 2.3.2.1.2

Raise $7$ to the power of $2$ .

$f' (7) = 14 - \frac{240}{49}$

Step 2.3.2.2

To write $14$ as a fraction with a common denominator, multiply by $\frac{49}{49}$ .

$f' (7) = 14 \cdot \frac{49}{49} - \frac{240}{49}$

Step 2.3.2.3

Combine $14$ and $\frac{49}{49}$ .

$f' (7) = \frac{14 \cdot 49}{49} - \frac{240}{49}$

Step 2.3.2.4

Combine the numerators over the common denominator.

$f' (7) = \frac{14 \cdot 49 - 240}{49}$

Step 2.3.2.5

Simplify the numerator.

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Step 2.3.2.5.1

Multiply $14$ by $49$ .

$f' (7) = \frac{686 - 240}{49}$

Step 2.3.2.5.2

Subtract $240$ from $686$ .

$f' (7) = \frac{446}{49}$

Step 2.3.2.6

The final answer is $\frac{446}{49}$ .

$\frac{446}{49}$

Step 2.4

Since the first derivative changed signs from negative to positive around $x = 2 \sqrt[3]{15}$ , then $x = 2 \sqrt[3]{15}$ is a local minimum.

$x = 2 \sqrt[3]{15}$ is a local minimum

Step 3

Compare the $f (x)$ values found for each value of $x$ in order to determine the absolute maximum and minimum over the given interval. The maximum will occur at the highest $f (x)$ value and the minimum will occur at the lowest $f (x)$ value.

No absolute maximum

Absolute Minimum: $(2 \sqrt[3]{15}, 12 \sqrt[3]{225})$

Step 4