Calculus Examples

$f (x) = - 413 x^{- 2} + 0.000004 x$

Step 1

Find the first derivative of the function.

Tap for more steps...

Step 1.1

By the Sum Rule, the derivative of $- 413 x^{- 2} + 0.000004 x$ with respect to $x$ is $\frac{d}{d x} [- 413 x^{- 2}] + \frac{d}{d x} [0.000004 x]$ .

$\frac{d}{d x} [- 413 x^{- 2}] + \frac{d}{d x} [0.000004 x]$

Step 1.2

Evaluate $\frac{d}{d x} [- 413 x^{- 2}]$ .

Tap for more steps...

Step 1.2.1

Since $- 413$ is constant with respect to $x$ , the derivative of $- 413 x^{- 2}$ with respect to $x$ is $- 413 \frac{d}{d x} [x^{- 2}]$ .

$- 413 \frac{d}{d x} [x^{- 2}] + \frac{d}{d x} [0.000004 x]$

Step 1.2.2

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

$- 413 (- 2 x^{- 3}) + \frac{d}{d x} [0.000004 x]$

Step 1.2.3

Multiply $- 2$ by $- 413$ .

$826 x^{- 3} + \frac{d}{d x} [0.000004 x]$

Step 1.3

Evaluate $\frac{d}{d x} [0.000004 x]$ .

Tap for more steps...

Step 1.3.1

Since $0.000004$ is constant with respect to $x$ , the derivative of $0.000004 x$ with respect to $x$ is $0.000004 \frac{d}{d x} [x]$ .

$826 x^{- 3} + 0.000004 \frac{d}{d x} [x]$

Step 1.3.2

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

$826 x^{- 3} + 0.000004 \cdot 1$

Step 1.3.3

Multiply $0.000004$ by $1$ .

$826 x^{- 3} + 0.000004$

Step 1.4

Simplify each term.

Tap for more steps...

Step 1.4.1

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

$826 \frac{1}{x^{3}} + 0.000004$

Step 1.4.2

Combine $826$ and $\frac{1}{x^{3}}$ .

$\frac{826}{x^{3}} + 0.000004$

Step 2

Find the second derivative of the function.

Tap for more steps...

Step 2.1

By the Sum Rule, the derivative of $\frac{826}{x^{3}} + 0.000004$ with respect to $x$ is $\frac{d}{d x} [\frac{826}{x^{3}}] + \frac{d}{d x} [0.000004]$ .

$f'' (x) = \frac{d}{d x} (\frac{826}{x^{3}}) + \frac{d}{d x} (0.000004)$

Step 2.2

Evaluate $\frac{d}{d x} [\frac{826}{x^{3}}]$ .

Tap for more steps...

Step 2.2.1

Since $826$ is constant with respect to $x$ , the derivative of $\frac{826}{x^{3}}$ with respect to $x$ is $826 \frac{d}{d x} [\frac{1}{x^{3}}]$ .

$f'' (x) = 826 \frac{d}{d x} (\frac{1}{x^{3}}) + \frac{d}{d x} (0.000004)$

Step 2.2.2

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

$f'' (x) = 826 \frac{d}{d x} ({(x^{3})}^{- 1}) + \frac{d}{d x} (0.000004)$

Step 2.2.3

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 $g (x) = x^{3}$ .

Tap for more steps...

Step 2.2.3.1

To apply the Chain Rule, set $u$ as $x^{3}$ .

$f'' (x) = 826 (\frac{d}{d u} (u^{- 1}) \frac{d}{d x} (x^{3})) + \frac{d}{d x} (0.000004)$

Step 2.2.3.2

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

$f'' (x) = 826 (- u^{- 2} \frac{d}{d x} x^{3}) + \frac{d}{d x} (0.000004)$

Step 2.2.3.3

Replace all occurrences of $u$ with $x^{3}$ .

$f'' (x) = 826 (- {(x^{3})}^{- 2} \frac{d}{d x} x^{3}) + \frac{d}{d x} (0.000004)$

Step 2.2.4

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

$f'' (x) = 826 (- {(x^{3})}^{- 2} (3 x^{2})) + \frac{d}{d x} (0.000004)$

Step 2.2.5

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

Tap for more steps...

Step 2.2.5.1

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

$f'' (x) = 826 (- x^{3 \cdot - 2} (3 x^{2})) + \frac{d}{d x} (0.000004)$

Step 2.2.5.2

Multiply $3$ by $- 2$ .

$f'' (x) = 826 (- x^{- 6} (3 x^{2})) + \frac{d}{d x} (0.000004)$

Step 2.2.6

Multiply $3$ by $- 1$ .

$f'' (x) = 826 (- 3 x^{- 6} x^{2}) + \frac{d}{d x} (0.000004)$

Step 2.2.7

Multiply $x^{- 6}$ by $x^{2}$ by adding the exponents.

Tap for more steps...

Step 2.2.7.1

Move $x^{2}$ .

$f'' (x) = 826 (- 3 (x^{2} x^{- 6})) + \frac{d}{d x} (0.000004)$

Step 2.2.7.2

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

$f'' (x) = 826 (- 3 x^{2 - 6}) + \frac{d}{d x} (0.000004)$

Step 2.2.7.3

Subtract $6$ from $2$ .

$f'' (x) = 826 (- 3 x^{- 4}) + \frac{d}{d x} (0.000004)$

Step 2.2.8

Multiply $- 3$ by $826$ .

$f'' (x) = - 2478 x^{- 4} + \frac{d}{d x} (0.000004)$

Step 2.3

Since $0.000004$ is constant with respect to $x$ , the derivative of $0.000004$ with respect to $x$ is $0$ .

$f'' (x) = - 2478 x^{- 4} + 0$

Step 2.4

Simplify.

Tap for more steps...

Step 2.4.1

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

$f'' (x) = - 2478 \frac{1}{x^{4}} + 0$

Step 2.4.2

Combine terms.

Tap for more steps...

Step 2.4.2.1

Combine $- 2478$ and $\frac{1}{x^{4}}$ .

$f'' (x) = \frac{- 2478}{x^{4}} + 0$

Step 2.4.2.2

Move the negative in front of the fraction.

$f'' (x) = - \frac{2478}{x^{4}} + 0$

Step 2.4.2.3

Add $- \frac{2478}{x^{4}}$ and $0$ .

$f'' (x) = - \frac{2478}{x^{4}}$

Step 3

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

$\frac{826}{x^{3}} + 0.000004 = 0$

Step 4

Find the first derivative.

Tap for more steps...

Step 4.1

Find the first derivative.

Tap for more steps...

Step 4.1.1

By the Sum Rule, the derivative of $- 413 x^{- 2} + 0.000004 x$ with respect to $x$ is $\frac{d}{d x} [- 413 x^{- 2}] + \frac{d}{d x} [0.000004 x]$ .

$\frac{d}{d x} [- 413 x^{- 2}] + \frac{d}{d x} [0.000004 x]$

Step 4.1.2

Evaluate $\frac{d}{d x} [- 413 x^{- 2}]$ .

Tap for more steps...

Step 4.1.2.1

Since $- 413$ is constant with respect to $x$ , the derivative of $- 413 x^{- 2}$ with respect to $x$ is $- 413 \frac{d}{d x} [x^{- 2}]$ .

$- 413 \frac{d}{d x} [x^{- 2}] + \frac{d}{d x} [0.000004 x]$

Step 4.1.2.2

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

$- 413 (- 2 x^{- 3}) + \frac{d}{d x} [0.000004 x]$

Step 4.1.2.3

Multiply $- 2$ by $- 413$ .

$826 x^{- 3} + \frac{d}{d x} [0.000004 x]$

Step 4.1.3

Evaluate $\frac{d}{d x} [0.000004 x]$ .

Tap for more steps...

Step 4.1.3.1

Since $0.000004$ is constant with respect to $x$ , the derivative of $0.000004 x$ with respect to $x$ is $0.000004 \frac{d}{d x} [x]$ .

$826 x^{- 3} + 0.000004 \frac{d}{d x} [x]$

Step 4.1.3.2

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

$826 x^{- 3} + 0.000004 \cdot 1$

Step 4.1.3.3

Multiply $0.000004$ by $1$ .

$826 x^{- 3} + 0.000004$

Step 4.1.4

Simplify each term.

Tap for more steps...

Step 4.1.4.1

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

$826 \frac{1}{x^{3}} + 0.000004$

Step 4.1.4.2

Combine $826$ and $\frac{1}{x^{3}}$ .

$f' (x) = \frac{826}{x^{3}} + 0.000004$

Step 4.2

The first derivative of $f (x)$ with respect to $x$ is $\frac{826}{x^{3}} + 0.000004$ .

$\frac{826}{x^{3}} + 0.000004$

Step 5

Set the first derivative equal to $0$ then solve the equation $\frac{826}{x^{3}} + 0.000004 = 0$ .

Tap for more steps...

Step 5.1

Set the first derivative equal to $0$ .

$\frac{826}{x^{3}} + 0.000004 = 0$

Step 5.2

Subtract $0.000004$ from both sides of the equation.

$\frac{826}{x^{3}} = - 0.000004$

Step 5.3

Find the LCD of the terms in the equation.

Tap for more steps...

Step 5.3.1

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 5.3.2

The LCM of one and any expression is the expression.

$x^{3}$

Step 5.4

Multiply each term in $\frac{826}{x^{3}} = - 0.000004$ by $x^{3}$ to eliminate the fractions.

Tap for more steps...

Step 5.4.1

Multiply each term in $\frac{826}{x^{3}} = - 0.000004$ by $x^{3}$ .

$\frac{826}{x^{3}} x^{3} = - 0.000004 x^{3}$

Step 5.4.2

Simplify the left side.

Tap for more steps...

Step 5.4.2.1

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

Tap for more steps...

Step 5.4.2.1.1

Cancel the common factor.

$\frac{826}{x^{3}} x^{3} = - 0.000004 x^{3}$

Step 5.4.2.1.2

Rewrite the expression.

$826 = - 0.000004 x^{3}$

Step 5.5

Solve the equation.

Tap for more steps...

Step 5.5.1

Rewrite the equation as $- 0.000004 x^{3} = 826$ .

$- 0.000004 x^{3} = 826$

Step 5.5.2

Subtract $826$ from both sides of the equation.

$- 0.000004 x^{3} - 826 = 0$

Step 5.5.3

Factor $- 0.000004$ out of $- 0.000004 x^{3} - 826$ .

Tap for more steps...

Step 5.5.3.1

Factor $- 0.000004$ out of $- 0.000004 x^{3}$ .

$- 0.000004 x^{3} - 826 = 0$

Step 5.5.3.2

Factor $- 0.000004$ out of $- 826$ .

$- 0.000004 x^{3} - 0.000004 \cdot 206500000 = 0$

Step 5.5.3.3

Factor $- 0.000004$ out of $- 0.000004 (x^{3}) - 0.000004 (206500000)$ .

$- 0.000004 (x^{3} + 206500000) = 0$

Step 5.5.4

Divide each term in $- 0.000004 (x^{3} + 206500000) = 0$ by $- 0.000004$ and simplify.

Tap for more steps...

Step 5.5.4.1

Divide each term in $- 0.000004 (x^{3} + 206500000) = 0$ by $- 0.000004$ .

$\frac{- 0.000004 (x^{3} + 206500000)}{- 0.000004} = \frac{0}{- 0.000004}$

Step 5.5.4.2

Simplify the left side.

Tap for more steps...

Step 5.5.4.2.1

Cancel the common factor of $- 0.000004$ .

Tap for more steps...

Step 5.5.4.2.1.1

Cancel the common factor.

$\frac{- 0.000004 (x^{3} + 206500000)}{- 0.000004} = \frac{0}{- 0.000004}$

Step 5.5.4.2.1.2

Divide $x^{3} + 206500000$ by $1$ .

$x^{3} + 206500000 = \frac{0}{- 0.000004}$

Step 5.5.4.3

Simplify the right side.

Tap for more steps...

Step 5.5.4.3.1

Divide $0$ by $- 0.000004$ .

$x^{3} + 206500000 = 0$

Step 5.5.5

Subtract $206500000$ from both sides of the equation.

$x^{3} = - 206500000$

Step 5.5.6

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

$x = \sqrt[3]{- 206500000}$

Step 5.5.7

Simplify $\sqrt[3]{- 206500000}$ .

Tap for more steps...

Step 5.5.7.1

Rewrite $- 206500000$ as ${(- 50)}^{3} \cdot 1652$ .

Tap for more steps...

Step 5.5.7.1.1

Factor $- 125000$ out of $- 206500000$ .

$x = \sqrt[3]{- 125000 (1652)}$

Step 5.5.7.1.2

Rewrite $- 125000$ as ${(- 50)}^{3}$ .

$x = \sqrt[3]{{(- 50)}^{3} \cdot 1652}$

Step 5.5.7.2

Pull terms out from under the radical.

$x = - 50 \sqrt[3]{1652}$

Step 6

Find the values where the derivative is undefined.

Tap for more steps...

Step 6.1

Set the denominator in $\frac{826}{x^{3}}$ equal to $0$ to find where the expression is undefined.

$x^{3} = 0$

Step 6.2

Solve for $x$ .

Tap for more steps...

Step 6.2.1

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

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

Step 6.2.2

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

Tap for more steps...

Step 6.2.2.1

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

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

Step 6.2.2.2

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

$x = 0$

Step 7

Critical points to evaluate.

$x = - 50 \sqrt[3]{1652}$

Step 8

Evaluate the second derivative at $x = - 50 \sqrt[3]{1652}$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$- \frac{2478}{{(- 50 \sqrt[3]{1652})}^{4}}$

Step 9

Evaluate the second derivative.

Tap for more steps...

Step 9.1

Simplify the denominator.

Tap for more steps...

Step 9.1.1

Apply the product rule to $- 50 \sqrt[3]{1652}$ .

$- \frac{2478}{{(- 50)}^{4} {\sqrt[3]{1652}}^{4}}$

Step 9.1.2

Raise $- 50$ to the power of $4$ .

$- \frac{2478}{6250000 {\sqrt[3]{1652}}^{4}}$

Step 9.1.3

Rewrite ${\sqrt[3]{1652}}^{4}$ as $\sqrt[3]{1652^{4}}$ .

$- \frac{2478}{6250000 \sqrt[3]{1652^{4}}}$

Step 9.1.4

Raise $1652$ to the power of $4$ .

$- \frac{2478}{6250000 \sqrt[3]{7448008642816}}$

Step 9.1.5

Rewrite $7448008642816$ as $1652^{3} \cdot 1652$ .

Tap for more steps...

Step 9.1.5.1

Factor $4508479808$ out of $7448008642816$ .

$- \frac{2478}{6250000 \sqrt[3]{4508479808 (1652)}}$

Step 9.1.5.2

Rewrite $4508479808$ as $1652^{3}$ .

$- \frac{2478}{6250000 \sqrt[3]{1652^{3} \cdot 1652}}$

Step 9.1.6

Pull terms out from under the radical.

$- \frac{2478}{6250000 \cdot 1652 \sqrt[3]{1652}}$

Step 9.1.7

Multiply $6250000$ by $1652$ .

$- \frac{2478}{10325000000 \sqrt[3]{1652}}$

Step 9.2

Cancel the common factor of $2478$ and $10325000000$ .

Tap for more steps...

Step 9.2.1

Factor $826$ out of $2478$ .

$- \frac{826 (3)}{10325000000 \sqrt[3]{1652}}$

Step 9.2.2

Cancel the common factors.

Tap for more steps...

Step 9.2.2.1

Factor $826$ out of $10325000000 \sqrt[3]{1652}$ .

$- \frac{826 (3)}{826 (12500000 \sqrt[3]{1652})}$

Step 9.2.2.2

Cancel the common factor.

$- \frac{826 \cdot 3}{826 (12500000 \sqrt[3]{1652})}$

Step 9.2.2.3

Rewrite the expression.

$- \frac{3}{12500000 \sqrt[3]{1652}}$

Step 9.3

Multiply $\frac{3}{12500000 \sqrt[3]{1652}}$ by $\frac{{\sqrt[3]{1652}}^{2}}{{\sqrt[3]{1652}}^{2}}$ .

$- (\frac{3}{12500000 \sqrt[3]{1652}} \cdot \frac{{\sqrt[3]{1652}}^{2}}{{\sqrt[3]{1652}}^{2}})$

Step 9.4

Combine and simplify the denominator.

Tap for more steps...

Step 9.4.1

Multiply $\frac{3}{12500000 \sqrt[3]{1652}}$ by $\frac{{\sqrt[3]{1652}}^{2}}{{\sqrt[3]{1652}}^{2}}$ .

$- \frac{3 {\sqrt[3]{1652}}^{2}}{12500000 \sqrt[3]{1652} {\sqrt[3]{1652}}^{2}}$

Step 9.4.2

Move $\sqrt[3]{1652}$ .

$- \frac{3 {\sqrt[3]{1652}}^{2}}{12500000 (\sqrt[3]{1652} {\sqrt[3]{1652}}^{2})}$

Step 9.4.3

Raise $\sqrt[3]{1652}$ to the power of $1$ .

$- \frac{3 {\sqrt[3]{1652}}^{2}}{12500000 ({\sqrt[3]{1652}}^{1} {\sqrt[3]{1652}}^{2})}$

Step 9.4.4

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

$- \frac{3 {\sqrt[3]{1652}}^{2}}{12500000 {\sqrt[3]{1652}}^{1 + 2}}$

Step 9.4.5

Add $1$ and $2$ .

$- \frac{3 {\sqrt[3]{1652}}^{2}}{12500000 {\sqrt[3]{1652}}^{3}}$

Step 9.4.6

Rewrite ${\sqrt[3]{1652}}^{3}$ as $1652$ .

Tap for more steps...

Step 9.4.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{1652}$ as $1652^{\frac{1}{3}}$ .

$- \frac{3 {\sqrt[3]{1652}}^{2}}{12500000 {(1652^{\frac{1}{3}})}^{3}}$

Step 9.4.6.2

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

$- \frac{3 {\sqrt[3]{1652}}^{2}}{12500000 \cdot 1652^{\frac{1}{3} \cdot 3}}$

Step 9.4.6.3

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

$- \frac{3 {\sqrt[3]{1652}}^{2}}{12500000 \cdot 1652^{\frac{3}{3}}}$

Step 9.4.6.4

Cancel the common factor of $3$ .

Tap for more steps...

Step 9.4.6.4.1

Cancel the common factor.

$- \frac{3 {\sqrt[3]{1652}}^{2}}{12500000 \cdot 1652^{\frac{3}{3}}}$

Step 9.4.6.4.2

Rewrite the expression.

$- \frac{3 {\sqrt[3]{1652}}^{2}}{12500000 \cdot 1652^{1}}$

Step 9.4.6.5

Evaluate the exponent.

$- \frac{3 {\sqrt[3]{1652}}^{2}}{12500000 \cdot 1652}$

Step 9.5

Simplify the numerator.

Tap for more steps...

Step 9.5.1

Rewrite ${\sqrt[3]{1652}}^{2}$ as $\sqrt[3]{1652^{2}}$ .

$- \frac{3 \sqrt[3]{1652^{2}}}{12500000 \cdot 1652}$

Step 9.5.2

Raise $1652$ to the power of $2$ .

$- \frac{3 \sqrt[3]{2729104}}{12500000 \cdot 1652}$

Step 9.5.3

Rewrite $2729104$ as $2^{3} \cdot 341138$ .

Tap for more steps...

Step 9.5.3.1

Factor $8$ out of $2729104$ .

$- \frac{3 \sqrt[3]{8 (341138)}}{12500000 \cdot 1652}$

Step 9.5.3.2

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

$- \frac{3 \sqrt[3]{2^{3} \cdot 341138}}{12500000 \cdot 1652}$

Step 9.5.4

Pull terms out from under the radical.

$- \frac{3 \cdot 2 \sqrt[3]{341138}}{12500000 \cdot 1652}$

Step 9.5.5

Multiply $3$ by $2$ .

$- \frac{6 \sqrt[3]{341138}}{12500000 \cdot 1652}$

Step 9.6

Reduce the expression by cancelling the common factors.

Tap for more steps...

Step 9.6.1

Multiply $12500000$ by $1652$ .

$- \frac{6 \sqrt[3]{341138}}{20650000000}$

Step 9.6.2

Cancel the common factor of $6$ and $20650000000$ .

Tap for more steps...

Step 9.6.2.1

Factor $2$ out of $6 \sqrt[3]{341138}$ .

$- \frac{2 (3 \sqrt[3]{341138})}{20650000000}$

Step 9.6.2.2

Cancel the common factors.

Tap for more steps...

Step 9.6.2.2.1

Factor $2$ out of $20650000000$ .

$- \frac{2 (3 \sqrt[3]{341138})}{2 (10325000000)}$

Step 9.6.2.2.2

Cancel the common factor.

$- \frac{2 (3 \sqrt[3]{341138})}{2 \cdot 10325000000}$

Step 9.6.2.2.3

Rewrite the expression.

$- \frac{3 \sqrt[3]{341138}}{10325000000}$

Step 10

$x = - 50 \sqrt[3]{1652}$ is a local maximum because the value of the second derivative is negative. This is referred to as the second derivative test.

$x = - 50 \sqrt[3]{1652}$ is a local maximum

Step 11

Find the y-value when $x = - 50 \sqrt[3]{1652}$ .

Tap for more steps...

Step 11.1

Replace the variable $x$ with $- 50 \sqrt[3]{1652}$ in the expression.

$f (- 50 \sqrt[3]{1652}) = - 413 {(- 50 \sqrt[3]{1652})}^{- 2} + 0.000004 (- 50 \sqrt[3]{1652})$

Step 11.2

Simplify the result.

Tap for more steps...

Step 11.2.1

Simplify each term.

Tap for more steps...

Step 11.2.1.1

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

$f (- 50 \sqrt[3]{1652}) = - 413 \frac{1}{{(- 50 \sqrt[3]{1652})}^{2}} + 0.000004 (- 50 \sqrt[3]{1652})$

Step 11.2.1.2

Simplify the denominator.

Tap for more steps...

Step 11.2.1.2.1

Apply the product rule to $- 50 \sqrt[3]{1652}$ .

$f (- 50 \sqrt[3]{1652}) = - 413 \frac{1}{{(- 50)}^{2} {\sqrt[3]{1652}}^{2}} + 0.000004 (- 50 \sqrt[3]{1652})$

Step 11.2.1.2.2

Raise $- 50$ to the power of $2$ .

$f (- 50 \sqrt[3]{1652}) = - 413 \frac{1}{2500 {\sqrt[3]{1652}}^{2}} + 0.000004 (- 50 \sqrt[3]{1652})$

Step 11.2.1.2.3

Rewrite ${\sqrt[3]{1652}}^{2}$ as $\sqrt[3]{1652^{2}}$ .

$f (- 50 \sqrt[3]{1652}) = - 413 \frac{1}{2500 \sqrt[3]{1652^{2}}} + 0.000004 (- 50 \sqrt[3]{1652})$

Step 11.2.1.2.4

Raise $1652$ to the power of $2$ .

$f (- 50 \sqrt[3]{1652}) = - 413 \frac{1}{2500 \sqrt[3]{2729104}} + 0.000004 (- 50 \sqrt[3]{1652})$

Step 11.2.1.2.5

Rewrite $2729104$ as $2^{3} \cdot 341138$ .

Tap for more steps...

Step 11.2.1.2.5.1

Factor $8$ out of $2729104$ .

$f (- 50 \sqrt[3]{1652}) = - 413 \frac{1}{2500 \sqrt[3]{8 (341138)}} + 0.000004 (- 50 \sqrt[3]{1652})$

Step 11.2.1.2.5.2

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

$f (- 50 \sqrt[3]{1652}) = - 413 \frac{1}{2500 \sqrt[3]{2^{3} \cdot 341138}} + 0.000004 (- 50 \sqrt[3]{1652})$

Step 11.2.1.2.6

Pull terms out from under the radical.

$f (- 50 \sqrt[3]{1652}) = - 413 \frac{1}{2500 \cdot (2 \sqrt[3]{341138})} + 0.000004 (- 50 \sqrt[3]{1652})$

Step 11.2.1.2.7

Multiply $2500$ by $2$ .

$f (- 50 \sqrt[3]{1652}) = - 413 \frac{1}{5000 \sqrt[3]{341138}} + 0.000004 (- 50 \sqrt[3]{1652})$

Step 11.2.1.3

Multiply $\frac{1}{5000 \sqrt[3]{341138}}$ by $\frac{{\sqrt[3]{341138}}^{2}}{{\sqrt[3]{341138}}^{2}}$ .

$f (- 50 \sqrt[3]{1652}) = - 413 (\frac{1}{5000 \sqrt[3]{341138}} \cdot \frac{{\sqrt[3]{341138}}^{2}}{{\sqrt[3]{341138}}^{2}}) + 0.000004 (- 50 \sqrt[3]{1652})$

Step 11.2.1.4

Combine and simplify the denominator.

Tap for more steps...

Step 11.2.1.4.1

Multiply $\frac{1}{5000 \sqrt[3]{341138}}$ by $\frac{{\sqrt[3]{341138}}^{2}}{{\sqrt[3]{341138}}^{2}}$ .

$f (- 50 \sqrt[3]{1652}) = - 413 \frac{{\sqrt[3]{341138}}^{2}}{5000 \sqrt[3]{341138} {\sqrt[3]{341138}}^{2}} + 0.000004 (- 50 \sqrt[3]{1652})$

Step 11.2.1.4.2

Move $\sqrt[3]{341138}$ .

$f (- 50 \sqrt[3]{1652}) = - 413 \frac{{\sqrt[3]{341138}}^{2}}{5000 (\sqrt[3]{341138} {\sqrt[3]{341138}}^{2})} + 0.000004 (- 50 \sqrt[3]{1652})$

Step 11.2.1.4.3

Raise $\sqrt[3]{341138}$ to the power of $1$ .

$f (- 50 \sqrt[3]{1652}) = - 413 \frac{{\sqrt[3]{341138}}^{2}}{5000 (\sqrt[3]{341138} {\sqrt[3]{341138}}^{2})} + 0.000004 (- 50 \sqrt[3]{1652})$

Step 11.2.1.4.4

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

$f (- 50 \sqrt[3]{1652}) = - 413 \frac{{\sqrt[3]{341138}}^{2}}{5000 {\sqrt[3]{341138}}^{1 + 2}} + 0.000004 (- 50 \sqrt[3]{1652})$

Step 11.2.1.4.5

Add $1$ and $2$ .

$f (- 50 \sqrt[3]{1652}) = - 413 \frac{{\sqrt[3]{341138}}^{2}}{5000 {\sqrt[3]{341138}}^{3}} + 0.000004 (- 50 \sqrt[3]{1652})$

Step 11.2.1.4.6

Rewrite ${\sqrt[3]{341138}}^{3}$ as $341138$ .

Tap for more steps...

Step 11.2.1.4.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{341138}$ as $341138^{\frac{1}{3}}$ .

$f (- 50 \sqrt[3]{1652}) = - 413 \frac{{\sqrt[3]{341138}}^{2}}{5000 {(341138^{\frac{1}{3}})}^{3}} + 0.000004 (- 50 \sqrt[3]{1652})$

Step 11.2.1.4.6.2

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

$f (- 50 \sqrt[3]{1652}) = - 413 \frac{{\sqrt[3]{341138}}^{2}}{5000 \cdot 341138^{\frac{1}{3} \cdot 3}} + 0.000004 (- 50 \sqrt[3]{1652})$

Step 11.2.1.4.6.3

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

$f (- 50 \sqrt[3]{1652}) = - 413 \frac{{\sqrt[3]{341138}}^{2}}{5000 \cdot 341138^{\frac{3}{3}}} + 0.000004 (- 50 \sqrt[3]{1652})$

Step 11.2.1.4.6.4

Cancel the common factor of $3$ .

Tap for more steps...

Step 11.2.1.4.6.4.1

Cancel the common factor.

$f (- 50 \sqrt[3]{1652}) = - 413 \frac{{\sqrt[3]{341138}}^{2}}{5000 \cdot 341138^{\frac{3}{3}}} + 0.000004 (- 50 \sqrt[3]{1652})$

Step 11.2.1.4.6.4.2

Rewrite the expression.

$f (- 50 \sqrt[3]{1652}) = - 413 \frac{{\sqrt[3]{341138}}^{2}}{5000 \cdot 341138} + 0.000004 (- 50 \sqrt[3]{1652})$

Step 11.2.1.4.6.5

Evaluate the exponent.

$f (- 50 \sqrt[3]{1652}) = - 413 \frac{{\sqrt[3]{341138}}^{2}}{5000 \cdot 341138} + 0.000004 (- 50 \sqrt[3]{1652})$

Step 11.2.1.5

Simplify the numerator.

Tap for more steps...

Step 11.2.1.5.1

Rewrite ${\sqrt[3]{341138}}^{2}$ as $\sqrt[3]{341138^{2}}$ .

$f (- 50 \sqrt[3]{1652}) = - 413 \frac{\sqrt[3]{341138^{2}}}{5000 \cdot 341138} + 0.000004 (- 50 \sqrt[3]{1652})$

Step 11.2.1.5.2

Raise $341138$ to the power of $2$ .

$f (- 50 \sqrt[3]{1652}) = - 413 \frac{\sqrt[3]{116375135044}}{5000 \cdot 341138} + 0.000004 (- 50 \sqrt[3]{1652})$

Step 11.2.1.5.3

Rewrite $116375135044$ as $413^{3} \cdot 1652$ .

Tap for more steps...

Step 11.2.1.5.3.1

Factor $70444997$ out of $116375135044$ .

$f (- 50 \sqrt[3]{1652}) = - 413 \frac{\sqrt[3]{70444997 (1652)}}{5000 \cdot 341138} + 0.000004 (- 50 \sqrt[3]{1652})$

Step 11.2.1.5.3.2

Rewrite $70444997$ as $413^{3}$ .

$f (- 50 \sqrt[3]{1652}) = - 413 \frac{\sqrt[3]{413^{3} \cdot 1652}}{5000 \cdot 341138} + 0.000004 (- 50 \sqrt[3]{1652})$

Step 11.2.1.5.4

Pull terms out from under the radical.

$f (- 50 \sqrt[3]{1652}) = - 413 \frac{413 \sqrt[3]{1652}}{5000 \cdot 341138} + 0.000004 (- 50 \sqrt[3]{1652})$

Step 11.2.1.6

Multiply $5000$ by $341138$ .

$f (- 50 \sqrt[3]{1652}) = - 413 \frac{413 \sqrt[3]{1652}}{1705690000} + 0.000004 (- 50 \sqrt[3]{1652})$

Step 11.2.1.7

Cancel the common factor of $413$ .

Tap for more steps...

Step 11.2.1.7.1

Factor $413$ out of $- 413$ .

$f (- 50 \sqrt[3]{1652}) = 413 (- 1) (\frac{413 \sqrt[3]{1652}}{1705690000}) + 0.000004 (- 50 \sqrt[3]{1652})$

Step 11.2.1.7.2

Factor $413$ out of $1705690000$ .

$f (- 50 \sqrt[3]{1652}) = 413 \cdot (- 1 \frac{413 \sqrt[3]{1652}}{413 \cdot 4130000}) + 0.000004 (- 50 \sqrt[3]{1652})$

Step 11.2.1.7.3

Cancel the common factor.

$f (- 50 \sqrt[3]{1652}) = 413 \cdot (- 1 \frac{413 \sqrt[3]{1652}}{413 \cdot 4130000}) + 0.000004 (- 50 \sqrt[3]{1652})$

Step 11.2.1.7.4

Rewrite the expression.

$f (- 50 \sqrt[3]{1652}) = - 1 \frac{413 \sqrt[3]{1652}}{4130000} + 0.000004 (- 50 \sqrt[3]{1652})$

Step 11.2.1.8

Cancel the common factor of $413$ and $4130000$ .

Tap for more steps...

Step 11.2.1.8.1

Factor $413$ out of $413 \sqrt[3]{1652}$ .

$f (- 50 \sqrt[3]{1652}) = - 1 \frac{413 (\sqrt[3]{1652})}{4130000} + 0.000004 (- 50 \sqrt[3]{1652})$

Step 11.2.1.8.2

Cancel the common factors.

Tap for more steps...

Step 11.2.1.8.2.1

Factor $413$ out of $4130000$ .

$f (- 50 \sqrt[3]{1652}) = - 1 \frac{413 \sqrt[3]{1652}}{413 \cdot 10000} + 0.000004 (- 50 \sqrt[3]{1652})$

Step 11.2.1.8.2.2

Cancel the common factor.

$f (- 50 \sqrt[3]{1652}) = - 1 \frac{413 \sqrt[3]{1652}}{413 \cdot 10000} + 0.000004 (- 50 \sqrt[3]{1652})$

Step 11.2.1.8.2.3

Rewrite the expression.

$f (- 50 \sqrt[3]{1652}) = - 1 \frac{\sqrt[3]{1652}}{10000} + 0.000004 (- 50 \sqrt[3]{1652})$

Step 11.2.1.9

Rewrite $- 1 \frac{\sqrt[3]{1652}}{10000}$ as $- \frac{\sqrt[3]{1652}}{10000}$ .

$f (- 50 \sqrt[3]{1652}) = - \frac{\sqrt[3]{1652}}{10000} + 0.000004 (- 50 \sqrt[3]{1652})$

Step 11.2.1.10

Multiply $0.000004 (- 50 \sqrt[3]{1652})$ .

Tap for more steps...

Step 11.2.1.10.1

Multiply $- 50$ by $0.000004$ .

$f (- 50 \sqrt[3]{1652}) = - \frac{\sqrt[3]{1652}}{10000} - 0.0002 \sqrt[3]{1652}$

Step 11.2.1.10.2

Multiply $- 0.0002$ by $\sqrt[3]{1652}$ .

$f (- 50 \sqrt[3]{1652}) = - \frac{\sqrt[3]{1652}}{10000} - 0.00236428$

Step 11.2.2

Subtract $0.00236428$ from $- \frac{\sqrt[3]{1652}}{10000}$ .

$f (- 50 \sqrt[3]{1652}) = - 0.00354642$

Step 11.2.3

The final answer is $- 0.00354642$ .

$y = - 0.00354642$

Step 12

These are the local extrema for $f (x) = - 413 x^{- 2} + 0.000004 x$ .

$(- 50 \sqrt[3]{1652}, - 0.00354642)$ is a local maxima

Step 13