Calculus Examples

$f (x) = c x^{2} - 5 x^{- 2}$

Step 1

Find the second derivative.

Tap for more steps...

Step 1.1

Find the first derivative.

Tap for more steps...

Step 1.1.1

By the Sum Rule, the derivative of $c x^{2} - 5 x^{- 2}$ with respect to $x$ is $\frac{d}{d x} [c x^{2}] + \frac{d}{d x} [- 5 x^{- 2}]$ .

$\frac{d}{d x} [c x^{2}] + \frac{d}{d x} [- 5 x^{- 2}]$

Step 1.1.2

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

Tap for more steps...

Step 1.1.2.1

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

$c \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 5 x^{- 2}]$

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 $n = 2$ .

$c (2 x) + \frac{d}{d x} [- 5 x^{- 2}]$

Step 1.1.2.3

Move $2$ to the left of $c$ .

$2 c x + \frac{d}{d x} [- 5 x^{- 2}]$

Step 1.1.3

Evaluate $\frac{d}{d x} [- 5 x^{- 2}]$ .

Tap for more steps...

Step 1.1.3.1

Since $- 5$ is constant with respect to $x$ , the derivative of $- 5 x^{- 2}$ with respect to $x$ is $- 5 \frac{d}{d x} [x^{- 2}]$ .

$2 c x - 5 \frac{d}{d x} [x^{- 2}]$

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 $n = - 2$ .

$2 c x - 5 (- 2 x^{- 3})$

Step 1.1.3.3

Multiply $- 2$ by $- 5$ .

$2 c x + 10 x^{- 3}$

Step 1.1.4

Simplify.

Tap for more steps...

Step 1.1.4.1

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

$2 c x + 10 \frac{1}{x^{3}}$

Step 1.1.4.2

Combine $10$ and $\frac{1}{x^{3}}$ .

$f' (x) = 2 c x + \frac{10}{x^{3}}$

Step 1.2

Find the second derivative.

Tap for more steps...

Step 1.2.1

By the Sum Rule, the derivative of $2 c x + \frac{10}{x^{3}}$ with respect to $x$ is $\frac{d}{d x} [2 c x] + \frac{d}{d x} [\frac{10}{x^{3}}]$ .

$\frac{d}{d x} [2 c x] + \frac{d}{d x} [\frac{10}{x^{3}}]$

Step 1.2.2

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

Tap for more steps...

Step 1.2.2.1

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

$2 c \frac{d}{d x} [x] + \frac{d}{d x} [\frac{10}{x^{3}}]$

Step 1.2.2.2

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

$2 c \cdot 1 + \frac{d}{d x} [\frac{10}{x^{3}}]$

Step 1.2.2.3

Multiply $2$ by $1$ .

$2 c + \frac{d}{d x} [\frac{10}{x^{3}}]$

Step 1.2.3

Evaluate $\frac{d}{d x} [\frac{10}{x^{3}}]$ .

Tap for more steps...

Step 1.2.3.1

Since $10$ is constant with respect to $x$ , the derivative of $\frac{10}{x^{3}}$ with respect to $x$ is $10 \frac{d}{d x} [\frac{1}{x^{3}}]$ .

$2 c + 10 \frac{d}{d x} [\frac{1}{x^{3}}]$

Step 1.2.3.2

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

$2 c + 10 \frac{d}{d x} [{(x^{3})}^{- 1}]$

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

Tap for more steps...

Step 1.2.3.3.1

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

$2 c + 10 (\frac{d}{d u} [u^{- 1}] \frac{d}{d x} [x^{3}])$

Step 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 $n = - 1$ .

$2 c + 10 (- u^{- 2} \frac{d}{d x} [x^{3}])$

Step 1.2.3.3.3

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

$2 c + 10 (- {(x^{3})}^{- 2} \frac{d}{d x} [x^{3}])$

Step 1.2.3.4

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

$2 c + 10 (- {(x^{3})}^{- 2} (3 x^{2}))$

Step 1.2.3.5

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

Tap for more steps...

Step 1.2.3.5.1

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

$2 c + 10 (- x^{3 \cdot - 2} (3 x^{2}))$

Step 1.2.3.5.2

Multiply $3$ by $- 2$ .

$2 c + 10 (- x^{- 6} (3 x^{2}))$

Step 1.2.3.6

Multiply $3$ by $- 1$ .

$2 c + 10 (- 3 x^{- 6} x^{2})$

Step 1.2.3.7

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

Tap for more steps...

Step 1.2.3.7.1

Move $x^{2}$ .

$2 c + 10 (- 3 (x^{2} x^{- 6}))$

Step 1.2.3.7.2

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

$2 c + 10 (- 3 x^{2 - 6})$

Step 1.2.3.7.3

Subtract $6$ from $2$ .

$2 c + 10 (- 3 x^{- 4})$

Step 1.2.3.8

Multiply $- 3$ by $10$ .

$2 c - 30 x^{- 4}$

Step 1.2.4

Simplify.

Tap for more steps...

Step 1.2.4.1

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

$2 c - 30 \frac{1}{x^{4}}$

Step 1.2.4.2

Combine terms.

Tap for more steps...

Step 1.2.4.2.1

Combine $- 30$ and $\frac{1}{x^{4}}$ .

$2 c + \frac{- 30}{x^{4}}$

Step 1.2.4.2.2

Move the negative in front of the fraction.

$f'' (x) = 2 c - \frac{30}{x^{4}}$

Step 1.3

The second derivative of $f (x)$ with respect to $x$ is $2 c - \frac{30}{x^{4}}$ .

$2 c - \frac{30}{x^{4}}$

Step 2

Set the second derivative equal to $0$ then solve the equation $2 c - \frac{30}{x^{4}} = 0$ .

Tap for more steps...

Step 2.1

Set the second derivative equal to $0$ .

$2 c - \frac{30}{x^{4}} = 0$

Step 2.2

Subtract $2 c$ from both sides of the equation.

$- \frac{30}{x^{4}} = - 2 c$

Step 2.3

Find the LCD of the terms in the equation.

Tap for more steps...

Step 2.3.1

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

$x^{4}, 1$

Step 2.3.2

The LCM of one and any expression is the expression.

$x^{4}$

Step 2.4

Multiply each term in $- \frac{30}{x^{4}} = - 2 c$ by $x^{4}$ to eliminate the fractions.

Tap for more steps...

Step 2.4.1

Multiply each term in $- \frac{30}{x^{4}} = - 2 c$ by $x^{4}$ .

$- \frac{30}{x^{4}} x^{4} = - 2 c x^{4}$

Step 2.4.2

Simplify the left side.

Tap for more steps...

Step 2.4.2.1

Cancel the common factor of $x^{4}$ .

Tap for more steps...

Step 2.4.2.1.1

Move the leading negative in $- \frac{30}{x^{4}}$ into the numerator.

$\frac{- 30}{x^{4}} x^{4} = - 2 c x^{4}$

Step 2.4.2.1.2

Cancel the common factor.

$\frac{- 30}{x^{4}} x^{4} = - 2 c x^{4}$

Step 2.4.2.1.3

Rewrite the expression.

$- 30 = - 2 c x^{4}$

Step 2.5

Solve the equation.

Tap for more steps...

Step 2.5.1

Rewrite the equation as $- 2 c x^{4} = - 30$ .

$- 2 c x^{4} = - 30$

Step 2.5.2

Divide each term in $- 2 c x^{4} = - 30$ by $- 2 c$ and simplify.

Tap for more steps...

Step 2.5.2.1

Divide each term in $- 2 c x^{4} = - 30$ by $- 2 c$ .

$\frac{- 2 c x^{4}}{- 2 c} = \frac{- 30}{- 2 c}$

Step 2.5.2.2

Simplify the left side.

Tap for more steps...

Step 2.5.2.2.1

Cancel the common factor of $- 2$ .

Tap for more steps...

Step 2.5.2.2.1.1

Cancel the common factor.

$\frac{- 2 c x^{4}}{- 2 c} = \frac{- 30}{- 2 c}$

Step 2.5.2.2.1.2

Rewrite the expression.

$\frac{c x^{4}}{c} = \frac{- 30}{- 2 c}$

Step 2.5.2.2.2

Cancel the common factor of $c$ .

Tap for more steps...

Step 2.5.2.2.2.1

Cancel the common factor.

$\frac{c x^{4}}{c} = \frac{- 30}{- 2 c}$

Step 2.5.2.2.2.2

Divide $x^{4}$ by $1$ .

$x^{4} = \frac{- 30}{- 2 c}$

Step 2.5.2.3

Simplify the right side.

Tap for more steps...

Step 2.5.2.3.1

Cancel the common factor of $- 30$ and $- 2$ .

Tap for more steps...

Step 2.5.2.3.1.1

Factor $- 2$ out of $- 30$ .

$x^{4} = \frac{- 2 (15)}{- 2 c}$

Step 2.5.2.3.1.2

Cancel the common factors.

Tap for more steps...

Step 2.5.2.3.1.2.1

Factor $- 2$ out of $- 2 c$ .

$x^{4} = \frac{- 2 (15)}{- 2 (c)}$

Step 2.5.2.3.1.2.2

Cancel the common factor.

$x^{4} = \frac{- 2 \cdot 15}{- 2 c}$

Step 2.5.2.3.1.2.3

Rewrite the expression.

$x^{4} = \frac{15}{c}$

Step 2.5.3

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

$x = \pm \sqrt[4]{\frac{15}{c}}$

Step 2.5.4

Simplify $\pm \sqrt[4]{\frac{15}{c}}$ .

Tap for more steps...

Step 2.5.4.1

Rewrite $\sqrt[4]{\frac{15}{c}}$ as $\frac{\sqrt[4]{15}}{\sqrt[4]{c}}$ .

$x = \pm \frac{\sqrt[4]{15}}{\sqrt[4]{c}}$

Step 2.5.4.2

Multiply $\frac{\sqrt[4]{15}}{\sqrt[4]{c}}$ by $\frac{{\sqrt[4]{c}}^{3}}{{\sqrt[4]{c}}^{3}}$ .

$x = \pm \frac{\sqrt[4]{15}}{\sqrt[4]{c}} \cdot \frac{{\sqrt[4]{c}}^{3}}{{\sqrt[4]{c}}^{3}}$

Step 2.5.4.3

Combine and simplify the denominator.

Tap for more steps...

Step 2.5.4.3.1

Multiply $\frac{\sqrt[4]{15}}{\sqrt[4]{c}}$ by $\frac{{\sqrt[4]{c}}^{3}}{{\sqrt[4]{c}}^{3}}$ .

$x = \pm \frac{\sqrt[4]{15} {\sqrt[4]{c}}^{3}}{\sqrt[4]{c} {\sqrt[4]{c}}^{3}}$

Step 2.5.4.3.2

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

$x = \pm \frac{\sqrt[4]{15} {\sqrt[4]{c}}^{3}}{{\sqrt[4]{c}}^{1} {\sqrt[4]{c}}^{3}}$

Step 2.5.4.3.3

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

$x = \pm \frac{\sqrt[4]{15} {\sqrt[4]{c}}^{3}}{{\sqrt[4]{c}}^{1 + 3}}$

Step 2.5.4.3.4

Add $1$ and $3$ .

$x = \pm \frac{\sqrt[4]{15} {\sqrt[4]{c}}^{3}}{{\sqrt[4]{c}}^{4}}$

Step 2.5.4.3.5

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

Tap for more steps...

Step 2.5.4.3.5.1

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

$x = \pm \frac{\sqrt[4]{15} {\sqrt[4]{c}}^{3}}{{(c^{\frac{1}{4}})}^{4}}$

Step 2.5.4.3.5.2

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

$x = \pm \frac{\sqrt[4]{15} {\sqrt[4]{c}}^{3}}{c^{\frac{1}{4} \cdot 4}}$

Step 2.5.4.3.5.3

Combine $\frac{1}{4}$ and $4$ .

$x = \pm \frac{\sqrt[4]{15} {\sqrt[4]{c}}^{3}}{c^{\frac{4}{4}}}$

Step 2.5.4.3.5.4

Cancel the common factor of $4$ .

Tap for more steps...

Step 2.5.4.3.5.4.1

Cancel the common factor.

$x = \pm \frac{\sqrt[4]{15} {\sqrt[4]{c}}^{3}}{c^{\frac{4}{4}}}$

Step 2.5.4.3.5.4.2

Rewrite the expression.

$x = \pm \frac{\sqrt[4]{15} {\sqrt[4]{c}}^{3}}{c^{1}}$

Step 2.5.4.3.5.5

Simplify.

$x = \pm \frac{\sqrt[4]{15} {\sqrt[4]{c}}^{3}}{c}$

Step 2.5.4.4

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

$x = \pm \frac{\sqrt[4]{15} \sqrt[4]{c^{3}}}{c}$

Step 2.5.4.5

Combine using the product rule for radicals.

$x = \pm \frac{\sqrt[4]{15 c^{3}}}{c}$

Step 2.5.5

The complete solution is the result of both the positive and negative portions of the solution.

Tap for more steps...

Step 2.5.5.1

First, use the positive value of the $\pm$ to find the first solution.

$x = \frac{\sqrt[4]{15 c^{3}}}{c}$

Step 2.5.5.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - \frac{\sqrt[4]{15 c^{3}}}{c}$

Step 2.5.5.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = \frac{\sqrt[4]{15 c^{3}}}{c}$

$x = - \frac{\sqrt[4]{15 c^{3}}}{c}$

$x = \frac{\sqrt[4]{15 c^{3}}}{c}$

$x = - \frac{\sqrt[4]{15 c^{3}}}{c}$

$x = \frac{\sqrt[4]{15 c^{3}}}{c}$

$x = - \frac{\sqrt[4]{15 c^{3}}}{c}$

$x = \frac{\sqrt[4]{15 c^{3}}}{c}$

$x = - \frac{\sqrt[4]{15 c^{3}}}{c}$

Step 3

Find the points where the second derivative is $0$ .

Tap for more steps...

Step 3.1

Substitute $\frac{\sqrt[4]{15 c^{3}}}{c}$ in $f (x) = c x^{2} - 5 x^{- 2}$ to find the value of $y$ .

Tap for more steps...

Step 3.1.1

Replace the variable $x$ with $\frac{\sqrt[4]{15 c^{3}}}{c}$ in the expression.

$f (\frac{\sqrt[4]{15 c^{3}}}{c}) = c {(\frac{\sqrt[4]{15 c^{3}}}{c})}^{2} - 5 {(\frac{\sqrt[4]{15 c^{3}}}{c})}^{- 2}$

Step 3.1.2

Simplify the result.

Tap for more steps...

Step 3.1.2.1

Simplify each term.

Tap for more steps...

Step 3.1.2.1.1

Apply the product rule to $\frac{\sqrt[4]{15 c^{3}}}{c}$ .

$f (\frac{\sqrt[4]{15 c^{3}}}{c}) = c (\frac{{\sqrt[4]{15 c^{3}}}^{2}}{c^{2}}) - 5 {(\frac{\sqrt[4]{15 c^{3}}}{c})}^{- 2}$

Step 3.1.2.1.2

Simplify the numerator.

Tap for more steps...

Step 3.1.2.1.2.1

Rewrite ${\sqrt[4]{15 c^{3}}}^{2}$ as $\sqrt{15 c^{3}}$ .

Tap for more steps...

Step 3.1.2.1.2.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{15 c^{3}}$ as ${(15 c^{3})}^{\frac{1}{4}}$ .

$f (\frac{\sqrt[4]{15 c^{3}}}{c}) = c (\frac{{({(15 c^{3})}^{\frac{1}{4}})}^{2}}{c^{2}}) - 5 {(\frac{\sqrt[4]{15 c^{3}}}{c})}^{- 2}$

Step 3.1.2.1.2.1.2

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

$f (\frac{\sqrt[4]{15 c^{3}}}{c}) = c (\frac{{(15 c^{3})}^{\frac{1}{4} \cdot 2}}{c^{2}}) - 5 {(\frac{\sqrt[4]{15 c^{3}}}{c})}^{- 2}$

Step 3.1.2.1.2.1.3

Combine $\frac{1}{4}$ and $2$ .

$f (\frac{\sqrt[4]{15 c^{3}}}{c}) = c (\frac{{(15 c^{3})}^{\frac{2}{4}}}{c^{2}}) - 5 {(\frac{\sqrt[4]{15 c^{3}}}{c})}^{- 2}$

Step 3.1.2.1.2.1.4

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

Tap for more steps...

Step 3.1.2.1.2.1.4.1

Factor $2$ out of $2$ .

$f (\frac{\sqrt[4]{15 c^{3}}}{c}) = c (\frac{{(15 c^{3})}^{\frac{2 (1)}{4}}}{c^{2}}) - 5 {(\frac{\sqrt[4]{15 c^{3}}}{c})}^{- 2}$

Step 3.1.2.1.2.1.4.2

Cancel the common factors.

Tap for more steps...

Step 3.1.2.1.2.1.4.2.1

Factor $2$ out of $4$ .

$f (\frac{\sqrt[4]{15 c^{3}}}{c}) = c (\frac{{(15 c^{3})}^{\frac{2 \cdot 1}{2 \cdot 2}}}{c^{2}}) - 5 {(\frac{\sqrt[4]{15 c^{3}}}{c})}^{- 2}$

Step 3.1.2.1.2.1.4.2.2

Cancel the common factor.

$f (\frac{\sqrt[4]{15 c^{3}}}{c}) = c (\frac{{(15 c^{3})}^{\frac{2 \cdot 1}{2 \cdot 2}}}{c^{2}}) - 5 {(\frac{\sqrt[4]{15 c^{3}}}{c})}^{- 2}$

Step 3.1.2.1.2.1.4.2.3

Rewrite the expression.

$f (\frac{\sqrt[4]{15 c^{3}}}{c}) = c (\frac{{(15 c^{3})}^{\frac{1}{2}}}{c^{2}}) - 5 {(\frac{\sqrt[4]{15 c^{3}}}{c})}^{- 2}$

Step 3.1.2.1.2.1.5

Rewrite ${(15 c^{3})}^{\frac{1}{2}}$ as $\sqrt{15 c^{3}}$ .

$f (\frac{\sqrt[4]{15 c^{3}}}{c}) = c (\frac{\sqrt{15 c^{3}}}{c^{2}}) - 5 {(\frac{\sqrt[4]{15 c^{3}}}{c})}^{- 2}$

Step 3.1.2.1.2.2

Rewrite $15 c^{3}$ as $c^{2} \cdot (15 c)$ .

Tap for more steps...

Step 3.1.2.1.2.2.1

Factor out $c^{2}$ .

$f (\frac{\sqrt[4]{15 c^{3}}}{c}) = c (\frac{\sqrt{15 (c^{2} c)}}{c^{2}}) - 5 {(\frac{\sqrt[4]{15 c^{3}}}{c})}^{- 2}$

Step 3.1.2.1.2.2.2

Reorder $15$ and $c^{2}$ .

$f (\frac{\sqrt[4]{15 c^{3}}}{c}) = c (\frac{\sqrt{c^{2} \cdot (15 c)}}{c^{2}}) - 5 {(\frac{\sqrt[4]{15 c^{3}}}{c})}^{- 2}$

Step 3.1.2.1.2.2.3

Add parentheses.

$f (\frac{\sqrt[4]{15 c^{3}}}{c}) = c (\frac{\sqrt{c^{2} \cdot (15 c)}}{c^{2}}) - 5 {(\frac{\sqrt[4]{15 c^{3}}}{c})}^{- 2}$

Step 3.1.2.1.2.3

Pull terms out from under the radical.

$f (\frac{\sqrt[4]{15 c^{3}}}{c}) = c (\frac{c \sqrt{15 c}}{c^{2}}) - 5 {(\frac{\sqrt[4]{15 c^{3}}}{c})}^{- 2}$

Step 3.1.2.1.3

Cancel the common factor of $c$ .

Tap for more steps...

Step 3.1.2.1.3.1

Factor $c$ out of $c^{2}$ .

$f (\frac{\sqrt[4]{15 c^{3}}}{c}) = c (\frac{c \sqrt{15 c}}{c \cdot c}) - 5 {(\frac{\sqrt[4]{15 c^{3}}}{c})}^{- 2}$

Step 3.1.2.1.3.2

Cancel the common factor.

$f (\frac{\sqrt[4]{15 c^{3}}}{c}) = c (\frac{c \sqrt{15 c}}{c \cdot c}) - 5 {(\frac{\sqrt[4]{15 c^{3}}}{c})}^{- 2}$

Step 3.1.2.1.3.3

Rewrite the expression.

$f (\frac{\sqrt[4]{15 c^{3}}}{c}) = \frac{c \sqrt{15 c}}{c} - 5 {(\frac{\sqrt[4]{15 c^{3}}}{c})}^{- 2}$

Step 3.1.2.1.4

Cancel the common factor of $c$ .

Tap for more steps...

Step 3.1.2.1.4.1

Cancel the common factor.

$f (\frac{\sqrt[4]{15 c^{3}}}{c}) = \frac{c \sqrt{15 c}}{c} - 5 {(\frac{\sqrt[4]{15 c^{3}}}{c})}^{- 2}$

Step 3.1.2.1.4.2

Divide $\sqrt{15 c}$ by $1$ .

$f (\frac{\sqrt[4]{15 c^{3}}}{c}) = \sqrt{15 c} - 5 {(\frac{\sqrt[4]{15 c^{3}}}{c})}^{- 2}$

Step 3.1.2.1.5

Change the sign of the exponent by rewriting the base as its reciprocal.

$f (\frac{\sqrt[4]{15 c^{3}}}{c}) = \sqrt{15 c} - 5 {(\frac{c}{\sqrt[4]{15 c^{3}}})}^{2}$

Step 3.1.2.1.6

Multiply $\frac{c}{\sqrt[4]{15 c^{3}}}$ by $\frac{{\sqrt[4]{15 c^{3}}}^{3}}{{\sqrt[4]{15 c^{3}}}^{3}}$ .

$f (\frac{\sqrt[4]{15 c^{3}}}{c}) = \sqrt{15 c} - 5 {(\frac{c}{\sqrt[4]{15 c^{3}}} \cdot \frac{{\sqrt[4]{15 c^{3}}}^{3}}{{\sqrt[4]{15 c^{3}}}^{3}})}^{2}$

Step 3.1.2.1.7

Combine and simplify the denominator.

Tap for more steps...

Step 3.1.2.1.7.1

Multiply $\frac{c}{\sqrt[4]{15 c^{3}}}$ by $\frac{{\sqrt[4]{15 c^{3}}}^{3}}{{\sqrt[4]{15 c^{3}}}^{3}}$ .

$f (\frac{\sqrt[4]{15 c^{3}}}{c}) = \sqrt{15 c} - 5 {(\frac{c {\sqrt[4]{15 c^{3}}}^{3}}{\sqrt[4]{15 c^{3}} {\sqrt[4]{15 c^{3}}}^{3}})}^{2}$

Step 3.1.2.1.7.2

Raise $\sqrt[4]{15 c^{3}}$ to the power of $1$ .

$f (\frac{\sqrt[4]{15 c^{3}}}{c}) = \sqrt{15 c} - 5 {(\frac{c {\sqrt[4]{15 c^{3}}}^{3}}{\sqrt[4]{15 c^{3}} {\sqrt[4]{15 c^{3}}}^{3}})}^{2}$

Step 3.1.2.1.7.3

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

$f (\frac{\sqrt[4]{15 c^{3}}}{c}) = \sqrt{15 c} - 5 {(\frac{c {\sqrt[4]{15 c^{3}}}^{3}}{{\sqrt[4]{15 c^{3}}}^{1 + 3}})}^{2}$

Step 3.1.2.1.7.4

Add $1$ and $3$ .

$f (\frac{\sqrt[4]{15 c^{3}}}{c}) = \sqrt{15 c} - 5 {(\frac{c {\sqrt[4]{15 c^{3}}}^{3}}{{\sqrt[4]{15 c^{3}}}^{4}})}^{2}$

Step 3.1.2.1.7.5

Rewrite ${\sqrt[4]{15 c^{3}}}^{4}$ as $15 c^{3}$ .

Tap for more steps...

Step 3.1.2.1.7.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{15 c^{3}}$ as ${(15 c^{3})}^{\frac{1}{4}}$ .

$f (\frac{\sqrt[4]{15 c^{3}}}{c}) = \sqrt{15 c} - 5 {(\frac{c {\sqrt[4]{15 c^{3}}}^{3}}{{({(15 c^{3})}^{\frac{1}{4}})}^{4}})}^{2}$

Step 3.1.2.1.7.5.2

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

$f (\frac{\sqrt[4]{15 c^{3}}}{c}) = \sqrt{15 c} - 5 {(\frac{c {\sqrt[4]{15 c^{3}}}^{3}}{{(15 c^{3})}^{\frac{1}{4} \cdot 4}})}^{2}$

Step 3.1.2.1.7.5.3

Combine $\frac{1}{4}$ and $4$ .

$f (\frac{\sqrt[4]{15 c^{3}}}{c}) = \sqrt{15 c} - 5 {(\frac{c {\sqrt[4]{15 c^{3}}}^{3}}{{(15 c^{3})}^{\frac{4}{4}}})}^{2}$

Step 3.1.2.1.7.5.4

Cancel the common factor of $4$ .

Tap for more steps...

Step 3.1.2.1.7.5.4.1

Cancel the common factor.

$f (\frac{\sqrt[4]{15 c^{3}}}{c}) = \sqrt{15 c} - 5 {(\frac{c {\sqrt[4]{15 c^{3}}}^{3}}{{(15 c^{3})}^{\frac{4}{4}}})}^{2}$

Step 3.1.2.1.7.5.4.2

Rewrite the expression.

$f (\frac{\sqrt[4]{15 c^{3}}}{c}) = \sqrt{15 c} - 5 {(\frac{c {\sqrt[4]{15 c^{3}}}^{3}}{15 c^{3}})}^{2}$

Step 3.1.2.1.7.5.5

Simplify.

$f (\frac{\sqrt[4]{15 c^{3}}}{c}) = \sqrt{15 c} - 5 {(\frac{c {\sqrt[4]{15 c^{3}}}^{3}}{15 c^{3}})}^{2}$

Step 3.1.2.1.8

Cancel the common factor of $c$ and $c^{3}$ .

Tap for more steps...

Step 3.1.2.1.8.1

Factor $c$ out of $c {\sqrt[4]{15 c^{3}}}^{3}$ .

$f (\frac{\sqrt[4]{15 c^{3}}}{c}) = \sqrt{15 c} - 5 {(\frac{c ({\sqrt[4]{15 c^{3}}}^{3})}{15 c^{3}})}^{2}$

Step 3.1.2.1.8.2

Cancel the common factors.

Tap for more steps...

Step 3.1.2.1.8.2.1

Factor $c$ out of $15 c^{3}$ .

$f (\frac{\sqrt[4]{15 c^{3}}}{c}) = \sqrt{15 c} - 5 {(\frac{c ({\sqrt[4]{15 c^{3}}}^{3})}{c (15 c^{2})})}^{2}$

Step 3.1.2.1.8.2.2

Cancel the common factor.

$f (\frac{\sqrt[4]{15 c^{3}}}{c}) = \sqrt{15 c} - 5 {(\frac{c {\sqrt[4]{15 c^{3}}}^{3}}{c (15 c^{2})})}^{2}$

Step 3.1.2.1.8.2.3

Rewrite the expression.

$f (\frac{\sqrt[4]{15 c^{3}}}{c}) = \sqrt{15 c} - 5 {(\frac{{\sqrt[4]{15 c^{3}}}^{3}}{15 c^{2}})}^{2}$

Step 3.1.2.1.9

Simplify the numerator.

Tap for more steps...

Step 3.1.2.1.9.1

Rewrite ${\sqrt[4]{15 c^{3}}}^{3}$ as $\sqrt[4]{{(15 c^{3})}^{3}}$ .

$f (\frac{\sqrt[4]{15 c^{3}}}{c}) = \sqrt{15 c} - 5 {(\frac{\sqrt[4]{{(15 c^{3})}^{3}}}{15 c^{2}})}^{2}$

Step 3.1.2.1.9.2

Apply the product rule to $15 c^{3}$ .

$f (\frac{\sqrt[4]{15 c^{3}}}{c}) = \sqrt{15 c} - 5 {(\frac{\sqrt[4]{15^{3} {(c^{3})}^{3}}}{15 c^{2}})}^{2}$

Step 3.1.2.1.9.3

Raise $15$ to the power of $3$ .

$f (\frac{\sqrt[4]{15 c^{3}}}{c}) = \sqrt{15 c} - 5 {(\frac{\sqrt[4]{3375 {(c^{3})}^{3}}}{15 c^{2}})}^{2}$

Step 3.1.2.1.9.4

Multiply the exponents in ${(c^{3})}^{3}$ .

Tap for more steps...

Step 3.1.2.1.9.4.1

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

$f (\frac{\sqrt[4]{15 c^{3}}}{c}) = \sqrt{15 c} - 5 {(\frac{\sqrt[4]{3375 c^{3 \cdot 3}}}{15 c^{2}})}^{2}$

Step 3.1.2.1.9.4.2

Multiply $3$ by $3$ .

$f (\frac{\sqrt[4]{15 c^{3}}}{c}) = \sqrt{15 c} - 5 {(\frac{\sqrt[4]{3375 c^{9}}}{15 c^{2}})}^{2}$

Step 3.1.2.1.9.5

Rewrite $3375 c^{9}$ as ${(c^{2})}^{4} \cdot (3375 c)$ .

Tap for more steps...

Step 3.1.2.1.9.5.1

Factor out $c^{8}$ .

$f (\frac{\sqrt[4]{15 c^{3}}}{c}) = \sqrt{15 c} - 5 {(\frac{\sqrt[4]{3375 (c^{8} c)}}{15 c^{2}})}^{2}$

Step 3.1.2.1.9.5.2

Rewrite $c^{8}$ as ${(c^{2})}^{4}$ .

$f (\frac{\sqrt[4]{15 c^{3}}}{c}) = \sqrt{15 c} - 5 {(\frac{\sqrt[4]{3375 ({(c^{2})}^{4} c)}}{15 c^{2}})}^{2}$

Step 3.1.2.1.9.5.3

Reorder $3375$ and ${(c^{2})}^{4}$ .

$f (\frac{\sqrt[4]{15 c^{3}}}{c}) = \sqrt{15 c} - 5 {(\frac{\sqrt[4]{{(c^{2})}^{4} \cdot (3375 c)}}{15 c^{2}})}^{2}$

Step 3.1.2.1.9.5.4

Add parentheses.

$f (\frac{\sqrt[4]{15 c^{3}}}{c}) = \sqrt{15 c} - 5 {(\frac{\sqrt[4]{{(c^{2})}^{4} \cdot (3375 c)}}{15 c^{2}})}^{2}$

Step 3.1.2.1.9.6

Pull terms out from under the radical.

$f (\frac{\sqrt[4]{15 c^{3}}}{c}) = \sqrt{15 c} - 5 {(\frac{c^{2} \sqrt[4]{3375 c}}{15 c^{2}})}^{2}$

Step 3.1.2.1.10

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

Tap for more steps...

Step 3.1.2.1.10.1

Cancel the common factor.

$f (\frac{\sqrt[4]{15 c^{3}}}{c}) = \sqrt{15 c} - 5 {(\frac{c^{2} \sqrt[4]{3375 c}}{15 c^{2}})}^{2}$

Step 3.1.2.1.10.2

Rewrite the expression.

$f (\frac{\sqrt[4]{15 c^{3}}}{c}) = \sqrt{15 c} - 5 {(\frac{\sqrt[4]{3375 c}}{15})}^{2}$

Step 3.1.2.1.11

Apply the product rule to $\frac{\sqrt[4]{3375 c}}{15}$ .

$f (\frac{\sqrt[4]{15 c^{3}}}{c}) = \sqrt{15 c} - 5 \frac{{\sqrt[4]{3375 c}}^{2}}{15^{2}}$

Step 3.1.2.1.12

Simplify the numerator.

Tap for more steps...

Step 3.1.2.1.12.1

Rewrite ${\sqrt[4]{3375 c}}^{2}$ as $\sqrt{3375 c}$ .

Tap for more steps...

Step 3.1.2.1.12.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{3375 c}$ as ${(3375 c)}^{\frac{1}{4}}$ .

$f (\frac{\sqrt[4]{15 c^{3}}}{c}) = \sqrt{15 c} - 5 \frac{{({(3375 c)}^{\frac{1}{4}})}^{2}}{15^{2}}$

Step 3.1.2.1.12.1.2

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

$f (\frac{\sqrt[4]{15 c^{3}}}{c}) = \sqrt{15 c} - 5 \frac{{(3375 c)}^{\frac{1}{4} \cdot 2}}{15^{2}}$

Step 3.1.2.1.12.1.3

Combine $\frac{1}{4}$ and $2$ .

$f (\frac{\sqrt[4]{15 c^{3}}}{c}) = \sqrt{15 c} - 5 \frac{{(3375 c)}^{\frac{2}{4}}}{15^{2}}$

Step 3.1.2.1.12.1.4

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

Tap for more steps...

Step 3.1.2.1.12.1.4.1

Factor $2$ out of $2$ .

$f (\frac{\sqrt[4]{15 c^{3}}}{c}) = \sqrt{15 c} - 5 \frac{{(3375 c)}^{\frac{2 (1)}{4}}}{15^{2}}$

Step 3.1.2.1.12.1.4.2

Cancel the common factors.

Tap for more steps...

Step 3.1.2.1.12.1.4.2.1

Factor $2$ out of $4$ .

$f (\frac{\sqrt[4]{15 c^{3}}}{c}) = \sqrt{15 c} - 5 \frac{{(3375 c)}^{\frac{2 \cdot 1}{2 \cdot 2}}}{15^{2}}$

Step 3.1.2.1.12.1.4.2.2

Cancel the common factor.

$f (\frac{\sqrt[4]{15 c^{3}}}{c}) = \sqrt{15 c} - 5 \frac{{(3375 c)}^{\frac{2 \cdot 1}{2 \cdot 2}}}{15^{2}}$

Step 3.1.2.1.12.1.4.2.3

Rewrite the expression.

$f (\frac{\sqrt[4]{15 c^{3}}}{c}) = \sqrt{15 c} - 5 \frac{{(3375 c)}^{\frac{1}{2}}}{15^{2}}$

Step 3.1.2.1.12.1.5

Rewrite ${(3375 c)}^{\frac{1}{2}}$ as $\sqrt{3375 c}$ .

$f (\frac{\sqrt[4]{15 c^{3}}}{c}) = \sqrt{15 c} - 5 \frac{\sqrt{3375 c}}{15^{2}}$

Step 3.1.2.1.12.2

Rewrite $3375 c$ as $15^{2} \cdot (15 c)$ .

Tap for more steps...

Step 3.1.2.1.12.2.1

Factor $225$ out of $3375$ .

$f (\frac{\sqrt[4]{15 c^{3}}}{c}) = \sqrt{15 c} - 5 \frac{\sqrt{225 (15) c}}{15^{2}}$

Step 3.1.2.1.12.2.2

Rewrite $225$ as $15^{2}$ .

$f (\frac{\sqrt[4]{15 c^{3}}}{c}) = \sqrt{15 c} - 5 \frac{\sqrt{15^{2} \cdot (15 c)}}{15^{2}}$

Step 3.1.2.1.12.2.3

Add parentheses.

$f (\frac{\sqrt[4]{15 c^{3}}}{c}) = \sqrt{15 c} - 5 \frac{\sqrt{15^{2} \cdot (15 c)}}{15^{2}}$

Step 3.1.2.1.12.3

Pull terms out from under the radical.

$f (\frac{\sqrt[4]{15 c^{3}}}{c}) = \sqrt{15 c} - 5 \frac{15 \sqrt{15 c}}{15^{2}}$

Step 3.1.2.1.13

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

$f (\frac{\sqrt[4]{15 c^{3}}}{c}) = \sqrt{15 c} - 5 \frac{15 \sqrt{15 c}}{225}$

Step 3.1.2.1.14

Cancel the common factor of $5$ .

Tap for more steps...

Step 3.1.2.1.14.1

Factor $5$ out of $- 5$ .

$f (\frac{\sqrt[4]{15 c^{3}}}{c}) = \sqrt{15 c} + 5 (- 1) (\frac{15 \sqrt{15 c}}{225})$

Step 3.1.2.1.14.2

Factor $5$ out of $225$ .

$f (\frac{\sqrt[4]{15 c^{3}}}{c}) = \sqrt{15 c} + 5 \cdot (- 1 \frac{15 \sqrt{15 c}}{5 \cdot 45})$

Step 3.1.2.1.14.3

Cancel the common factor.

$f (\frac{\sqrt[4]{15 c^{3}}}{c}) = \sqrt{15 c} + 5 \cdot (- 1 \frac{15 \sqrt{15 c}}{5 \cdot 45})$

Step 3.1.2.1.14.4

Rewrite the expression.

$f (\frac{\sqrt[4]{15 c^{3}}}{c}) = \sqrt{15 c} - 1 \frac{15 \sqrt{15 c}}{45}$

Step 3.1.2.1.15

Cancel the common factor of $15$ and $45$ .

Tap for more steps...

Step 3.1.2.1.15.1

Factor $15$ out of $15 \sqrt{15 c}$ .

$f (\frac{\sqrt[4]{15 c^{3}}}{c}) = \sqrt{15 c} - 1 \frac{15 (\sqrt{15 c})}{45}$

Step 3.1.2.1.15.2

Cancel the common factors.

Tap for more steps...

Step 3.1.2.1.15.2.1

Factor $15$ out of $45$ .

$f (\frac{\sqrt[4]{15 c^{3}}}{c}) = \sqrt{15 c} - 1 \frac{15 \sqrt{15 c}}{15 \cdot 3}$

Step 3.1.2.1.15.2.2

Cancel the common factor.

$f (\frac{\sqrt[4]{15 c^{3}}}{c}) = \sqrt{15 c} - 1 \frac{15 \sqrt{15 c}}{15 \cdot 3}$

Step 3.1.2.1.15.2.3

Rewrite the expression.

$f (\frac{\sqrt[4]{15 c^{3}}}{c}) = \sqrt{15 c} - 1 \frac{\sqrt{15 c}}{3}$

Step 3.1.2.1.16

Rewrite $- 1 \frac{\sqrt{15 c}}{3}$ as $- \frac{\sqrt{15 c}}{3}$ .

$f (\frac{\sqrt[4]{15 c^{3}}}{c}) = \sqrt{15 c} - \frac{\sqrt{15 c}}{3}$

Step 3.1.2.2

To write $\sqrt{15 c}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$f (\frac{\sqrt[4]{15 c^{3}}}{c}) = \sqrt{15 c} \cdot \frac{3}{3} - \frac{\sqrt{15 c}}{3}$

Step 3.1.2.3

Simplify terms.

Tap for more steps...

Step 3.1.2.3.1

Combine $\sqrt{15 c}$ and $\frac{3}{3}$ .

$f (\frac{\sqrt[4]{15 c^{3}}}{c}) = \frac{\sqrt{15 c} \cdot 3}{3} - \frac{\sqrt{15 c}}{3}$

Step 3.1.2.3.2

Combine the numerators over the common denominator.

$f (\frac{\sqrt[4]{15 c^{3}}}{c}) = \frac{\sqrt{15 c} \cdot 3 - \sqrt{15 c}}{3}$

Step 3.1.2.4

Simplify the numerator.

Tap for more steps...

Step 3.1.2.4.1

Move $3$ to the left of $\sqrt{15 c}$ .

$f (\frac{\sqrt[4]{15 c^{3}}}{c}) = \frac{3 \cdot \sqrt{15 c} - \sqrt{15 c}}{3}$

Step 3.1.2.4.2

Subtract $\sqrt{15 c}$ from $3 \sqrt{15 c}$ .

$f (\frac{\sqrt[4]{15 c^{3}}}{c}) = \frac{2 \sqrt{15 c}}{3}$

Step 3.1.2.5

The final answer is $\frac{2 \sqrt{15 c}}{3}$ .

$\frac{2 \sqrt{15 c}}{3}$

Step 3.2

The point found by substituting $\frac{\sqrt[4]{15 c^{3}}}{c}$ in $f (x) = c x^{2} - 5 x^{- 2}$ is $(\frac{\sqrt[4]{15 c^{3}}}{c}, \frac{2 \sqrt{15 c}}{3})$ . This point can be an inflection point.

$(\frac{\sqrt[4]{15 c^{3}}}{c}, \frac{2 \sqrt{15 c}}{3})$

Step 3.3

Substitute $- \frac{\sqrt[4]{15 c^{3}}}{c}$ in $f (x) = c x^{2} - 5 x^{- 2}$ to find the value of $y$ .

Tap for more steps...

Step 3.3.1

Replace the variable $x$ with $- \frac{\sqrt[4]{15 c^{3}}}{c}$ in the expression.

$f (- \frac{\sqrt[4]{15 c^{3}}}{c}) = c {(- \frac{\sqrt[4]{15 c^{3}}}{c})}^{2} - 5 {(- \frac{\sqrt[4]{15 c^{3}}}{c})}^{- 2}$

Step 3.3.2

Simplify the result.

Tap for more steps...

Step 3.3.2.1

Simplify each term.

Tap for more steps...

Step 3.3.2.1.1

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

Tap for more steps...

Step 3.3.2.1.1.1

Apply the product rule to $- \frac{\sqrt[4]{15 c^{3}}}{c}$ .

$f (- \frac{\sqrt[4]{15 c^{3}}}{c}) = c ({(- 1)}^{2} {(\frac{\sqrt[4]{15 c^{3}}}{c})}^{2}) - 5 {(- \frac{\sqrt[4]{15 c^{3}}}{c})}^{- 2}$

Step 3.3.2.1.1.2

Apply the product rule to $\frac{\sqrt[4]{15 c^{3}}}{c}$ .

$f (- \frac{\sqrt[4]{15 c^{3}}}{c}) = c ({(- 1)}^{2} (\frac{{\sqrt[4]{15 c^{3}}}^{2}}{c^{2}})) - 5 {(- \frac{\sqrt[4]{15 c^{3}}}{c})}^{- 2}$

Step 3.3.2.1.2

Rewrite using the commutative property of multiplication.

$f (- \frac{\sqrt[4]{15 c^{3}}}{c}) = {(- 1)}^{2} c (\frac{{\sqrt[4]{15 c^{3}}}^{2}}{c^{2}}) - 5 {(- \frac{\sqrt[4]{15 c^{3}}}{c})}^{- 2}$

Step 3.3.2.1.3

Cancel the common factor of $c$ .

Tap for more steps...

Step 3.3.2.1.3.1

Factor $c$ out of ${(- 1)}^{2} c$ .

$f (- \frac{\sqrt[4]{15 c^{3}}}{c}) = c {(- 1)}^{2} (\frac{{\sqrt[4]{15 c^{3}}}^{2}}{c^{2}}) - 5 {(- \frac{\sqrt[4]{15 c^{3}}}{c})}^{- 2}$

Step 3.3.2.1.3.2

Factor $c$ out of $c^{2}$ .

$f (- \frac{\sqrt[4]{15 c^{3}}}{c}) = c {(- 1)}^{2} (\frac{{\sqrt[4]{15 c^{3}}}^{2}}{c \cdot c}) - 5 {(- \frac{\sqrt[4]{15 c^{3}}}{c})}^{- 2}$

Step 3.3.2.1.3.3

Cancel the common factor.

$f (- \frac{\sqrt[4]{15 c^{3}}}{c}) = c {(- 1)}^{2} (\frac{{\sqrt[4]{15 c^{3}}}^{2}}{c \cdot c}) - 5 {(- \frac{\sqrt[4]{15 c^{3}}}{c})}^{- 2}$

Step 3.3.2.1.3.4

Rewrite the expression.

$f (- \frac{\sqrt[4]{15 c^{3}}}{c}) = {(- 1)}^{2} (\frac{{\sqrt[4]{15 c^{3}}}^{2}}{c}) - 5 {(- \frac{\sqrt[4]{15 c^{3}}}{c})}^{- 2}$

Step 3.3.2.1.4

Combine ${(- 1)}^{2}$ and $\frac{{\sqrt[4]{15 c^{3}}}^{2}}{c}$ .

$f (- \frac{\sqrt[4]{15 c^{3}}}{c}) = \frac{{(- 1)}^{2} {\sqrt[4]{15 c^{3}}}^{2}}{c} - 5 {(- \frac{\sqrt[4]{15 c^{3}}}{c})}^{- 2}$

Step 3.3.2.1.5

Simplify the numerator.

Tap for more steps...

Step 3.3.2.1.5.1

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

$f (- \frac{\sqrt[4]{15 c^{3}}}{c}) = \frac{1 {\sqrt[4]{15 c^{3}}}^{2}}{c} - 5 {(- \frac{\sqrt[4]{15 c^{3}}}{c})}^{- 2}$

Step 3.3.2.1.5.2

Rewrite ${\sqrt[4]{15 c^{3}}}^{2}$ as $\sqrt{15 c^{3}}$ .

Tap for more steps...

Step 3.3.2.1.5.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{15 c^{3}}$ as ${(15 c^{3})}^{\frac{1}{4}}$ .

$f (- \frac{\sqrt[4]{15 c^{3}}}{c}) = \frac{1 {({(15 c^{3})}^{\frac{1}{4}})}^{2}}{c} - 5 {(- \frac{\sqrt[4]{15 c^{3}}}{c})}^{- 2}$

Step 3.3.2.1.5.2.2

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

$f (- \frac{\sqrt[4]{15 c^{3}}}{c}) = \frac{1 {(15 c^{3})}^{\frac{1}{4} \cdot 2}}{c} - 5 {(- \frac{\sqrt[4]{15 c^{3}}}{c})}^{- 2}$

Step 3.3.2.1.5.2.3

Combine $\frac{1}{4}$ and $2$ .

$f (- \frac{\sqrt[4]{15 c^{3}}}{c}) = \frac{1 {(15 c^{3})}^{\frac{2}{4}}}{c} - 5 {(- \frac{\sqrt[4]{15 c^{3}}}{c})}^{- 2}$

Step 3.3.2.1.5.2.4

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

Tap for more steps...

Step 3.3.2.1.5.2.4.1

Factor $2$ out of $2$ .

$f (- \frac{\sqrt[4]{15 c^{3}}}{c}) = \frac{1 {(15 c^{3})}^{\frac{2 (1)}{4}}}{c} - 5 {(- \frac{\sqrt[4]{15 c^{3}}}{c})}^{- 2}$

Step 3.3.2.1.5.2.4.2

Cancel the common factors.

Tap for more steps...

Step 3.3.2.1.5.2.4.2.1

Factor $2$ out of $4$ .

$f (- \frac{\sqrt[4]{15 c^{3}}}{c}) = \frac{1 {(15 c^{3})}^{\frac{2 \cdot 1}{2 \cdot 2}}}{c} - 5 {(- \frac{\sqrt[4]{15 c^{3}}}{c})}^{- 2}$

Step 3.3.2.1.5.2.4.2.2

Cancel the common factor.

$f (- \frac{\sqrt[4]{15 c^{3}}}{c}) = \frac{1 {(15 c^{3})}^{\frac{2 \cdot 1}{2 \cdot 2}}}{c} - 5 {(- \frac{\sqrt[4]{15 c^{3}}}{c})}^{- 2}$

Step 3.3.2.1.5.2.4.2.3

Rewrite the expression.

$f (- \frac{\sqrt[4]{15 c^{3}}}{c}) = \frac{1 {(15 c^{3})}^{\frac{1}{2}}}{c} - 5 {(- \frac{\sqrt[4]{15 c^{3}}}{c})}^{- 2}$

Step 3.3.2.1.5.2.5

Rewrite ${(15 c^{3})}^{\frac{1}{2}}$ as $\sqrt{15 c^{3}}$ .

$f (- \frac{\sqrt[4]{15 c^{3}}}{c}) = \frac{1 \sqrt{15 c^{3}}}{c} - 5 {(- \frac{\sqrt[4]{15 c^{3}}}{c})}^{- 2}$

Step 3.3.2.1.5.3

Rewrite $15 c^{3}$ as $c^{2} \cdot (15 c)$ .

Tap for more steps...

Step 3.3.2.1.5.3.1

Factor out $c^{2}$ .

$f (- \frac{\sqrt[4]{15 c^{3}}}{c}) = \frac{1 \sqrt{15 (c^{2} c)}}{c} - 5 {(- \frac{\sqrt[4]{15 c^{3}}}{c})}^{- 2}$

Step 3.3.2.1.5.3.2

Reorder $15$ and $c^{2}$ .

$f (- \frac{\sqrt[4]{15 c^{3}}}{c}) = \frac{1 \sqrt{c^{2} \cdot (15 c)}}{c} - 5 {(- \frac{\sqrt[4]{15 c^{3}}}{c})}^{- 2}$

Step 3.3.2.1.5.3.3

Add parentheses.

$f (- \frac{\sqrt[4]{15 c^{3}}}{c}) = \frac{1 \sqrt{c^{2} \cdot (15 c)}}{c} - 5 {(- \frac{\sqrt[4]{15 c^{3}}}{c})}^{- 2}$

Step 3.3.2.1.5.4

Pull terms out from under the radical.

$f (- \frac{\sqrt[4]{15 c^{3}}}{c}) = \frac{1 (c \sqrt{15 c})}{c} - 5 {(- \frac{\sqrt[4]{15 c^{3}}}{c})}^{- 2}$

Step 3.3.2.1.5.5

Multiply $c \sqrt{15 c}$ by $1$ .

$f (- \frac{\sqrt[4]{15 c^{3}}}{c}) = \frac{c \sqrt{15 c}}{c} - 5 {(- \frac{\sqrt[4]{15 c^{3}}}{c})}^{- 2}$

Step 3.3.2.1.6

Cancel the common factor of $c$ .

Tap for more steps...

Step 3.3.2.1.6.1

Cancel the common factor.

$f (- \frac{\sqrt[4]{15 c^{3}}}{c}) = \frac{c \sqrt{15 c}}{c} - 5 {(- \frac{\sqrt[4]{15 c^{3}}}{c})}^{- 2}$

Step 3.3.2.1.6.2

Divide $\sqrt{15 c}$ by $1$ .

$f (- \frac{\sqrt[4]{15 c^{3}}}{c}) = \sqrt{15 c} - 5 {(- \frac{\sqrt[4]{15 c^{3}}}{c})}^{- 2}$

Step 3.3.2.1.7

Change the sign of the exponent by rewriting the base as its reciprocal.

$f (- \frac{\sqrt[4]{15 c^{3}}}{c}) = \sqrt{15 c} - 5 {(- \frac{c}{\sqrt[4]{15 c^{3}}})}^{2}$

Step 3.3.2.1.8

Multiply $\frac{c}{\sqrt[4]{15 c^{3}}}$ by $\frac{{\sqrt[4]{15 c^{3}}}^{3}}{{\sqrt[4]{15 c^{3}}}^{3}}$ .

$f (- \frac{\sqrt[4]{15 c^{3}}}{c}) = \sqrt{15 c} - 5 {(- (\frac{c}{\sqrt[4]{15 c^{3}}} \cdot \frac{{\sqrt[4]{15 c^{3}}}^{3}}{{\sqrt[4]{15 c^{3}}}^{3}}))}^{2}$

Step 3.3.2.1.9

Combine and simplify the denominator.

Tap for more steps...

Step 3.3.2.1.9.1

Multiply $\frac{c}{\sqrt[4]{15 c^{3}}}$ by $\frac{{\sqrt[4]{15 c^{3}}}^{3}}{{\sqrt[4]{15 c^{3}}}^{3}}$ .

$f (- \frac{\sqrt[4]{15 c^{3}}}{c}) = \sqrt{15 c} - 5 {(- \frac{c {\sqrt[4]{15 c^{3}}}^{3}}{\sqrt[4]{15 c^{3}} {\sqrt[4]{15 c^{3}}}^{3}})}^{2}$

Step 3.3.2.1.9.2

Raise $\sqrt[4]{15 c^{3}}$ to the power of $1$ .

$f (- \frac{\sqrt[4]{15 c^{3}}}{c}) = \sqrt{15 c} - 5 {(- \frac{c {\sqrt[4]{15 c^{3}}}^{3}}{\sqrt[4]{15 c^{3}} {\sqrt[4]{15 c^{3}}}^{3}})}^{2}$

Step 3.3.2.1.9.3

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

$f (- \frac{\sqrt[4]{15 c^{3}}}{c}) = \sqrt{15 c} - 5 {(- \frac{c {\sqrt[4]{15 c^{3}}}^{3}}{{\sqrt[4]{15 c^{3}}}^{1 + 3}})}^{2}$

Step 3.3.2.1.9.4

Add $1$ and $3$ .

$f (- \frac{\sqrt[4]{15 c^{3}}}{c}) = \sqrt{15 c} - 5 {(- \frac{c {\sqrt[4]{15 c^{3}}}^{3}}{{\sqrt[4]{15 c^{3}}}^{4}})}^{2}$

Step 3.3.2.1.9.5

Rewrite ${\sqrt[4]{15 c^{3}}}^{4}$ as $15 c^{3}$ .

Tap for more steps...

Step 3.3.2.1.9.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{15 c^{3}}$ as ${(15 c^{3})}^{\frac{1}{4}}$ .

$f (- \frac{\sqrt[4]{15 c^{3}}}{c}) = \sqrt{15 c} - 5 {(- \frac{c {\sqrt[4]{15 c^{3}}}^{3}}{{({(15 c^{3})}^{\frac{1}{4}})}^{4}})}^{2}$

Step 3.3.2.1.9.5.2

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

$f (- \frac{\sqrt[4]{15 c^{3}}}{c}) = \sqrt{15 c} - 5 {(- \frac{c {\sqrt[4]{15 c^{3}}}^{3}}{{(15 c^{3})}^{\frac{1}{4} \cdot 4}})}^{2}$

Step 3.3.2.1.9.5.3

Combine $\frac{1}{4}$ and $4$ .

$f (- \frac{\sqrt[4]{15 c^{3}}}{c}) = \sqrt{15 c} - 5 {(- \frac{c {\sqrt[4]{15 c^{3}}}^{3}}{{(15 c^{3})}^{\frac{4}{4}}})}^{2}$

Step 3.3.2.1.9.5.4

Cancel the common factor of $4$ .

Tap for more steps...

Step 3.3.2.1.9.5.4.1

Cancel the common factor.

$f (- \frac{\sqrt[4]{15 c^{3}}}{c}) = \sqrt{15 c} - 5 {(- \frac{c {\sqrt[4]{15 c^{3}}}^{3}}{{(15 c^{3})}^{\frac{4}{4}}})}^{2}$

Step 3.3.2.1.9.5.4.2

Rewrite the expression.

$f (- \frac{\sqrt[4]{15 c^{3}}}{c}) = \sqrt{15 c} - 5 {(- \frac{c {\sqrt[4]{15 c^{3}}}^{3}}{15 c^{3}})}^{2}$

Step 3.3.2.1.9.5.5

Simplify.

$f (- \frac{\sqrt[4]{15 c^{3}}}{c}) = \sqrt{15 c} - 5 {(- \frac{c {\sqrt[4]{15 c^{3}}}^{3}}{15 c^{3}})}^{2}$

Step 3.3.2.1.10

Cancel the common factor of $c$ and $c^{3}$ .

Tap for more steps...

Step 3.3.2.1.10.1

Factor $c$ out of $c {\sqrt[4]{15 c^{3}}}^{3}$ .

$f (- \frac{\sqrt[4]{15 c^{3}}}{c}) = \sqrt{15 c} - 5 {(- \frac{c ({\sqrt[4]{15 c^{3}}}^{3})}{15 c^{3}})}^{2}$

Step 3.3.2.1.10.2

Cancel the common factors.

Tap for more steps...

Step 3.3.2.1.10.2.1

Factor $c$ out of $15 c^{3}$ .

$f (- \frac{\sqrt[4]{15 c^{3}}}{c}) = \sqrt{15 c} - 5 {(- \frac{c ({\sqrt[4]{15 c^{3}}}^{3})}{c (15 c^{2})})}^{2}$

Step 3.3.2.1.10.2.2

Cancel the common factor.

$f (- \frac{\sqrt[4]{15 c^{3}}}{c}) = \sqrt{15 c} - 5 {(- \frac{c {\sqrt[4]{15 c^{3}}}^{3}}{c (15 c^{2})})}^{2}$

Step 3.3.2.1.10.2.3

Rewrite the expression.

$f (- \frac{\sqrt[4]{15 c^{3}}}{c}) = \sqrt{15 c} - 5 {(- \frac{{\sqrt[4]{15 c^{3}}}^{3}}{15 c^{2}})}^{2}$

Step 3.3.2.1.11

Simplify the numerator.

Tap for more steps...

Step 3.3.2.1.11.1

Rewrite ${\sqrt[4]{15 c^{3}}}^{3}$ as $\sqrt[4]{{(15 c^{3})}^{3}}$ .

$f (- \frac{\sqrt[4]{15 c^{3}}}{c}) = \sqrt{15 c} - 5 {(- \frac{\sqrt[4]{{(15 c^{3})}^{3}}}{15 c^{2}})}^{2}$

Step 3.3.2.1.11.2

Apply the product rule to $15 c^{3}$ .

$f (- \frac{\sqrt[4]{15 c^{3}}}{c}) = \sqrt{15 c} - 5 {(- \frac{\sqrt[4]{15^{3} {(c^{3})}^{3}}}{15 c^{2}})}^{2}$

Step 3.3.2.1.11.3

Raise $15$ to the power of $3$ .

$f (- \frac{\sqrt[4]{15 c^{3}}}{c}) = \sqrt{15 c} - 5 {(- \frac{\sqrt[4]{3375 {(c^{3})}^{3}}}{15 c^{2}})}^{2}$

Step 3.3.2.1.11.4

Multiply the exponents in ${(c^{3})}^{3}$ .

Tap for more steps...

Step 3.3.2.1.11.4.1

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

$f (- \frac{\sqrt[4]{15 c^{3}}}{c}) = \sqrt{15 c} - 5 {(- \frac{\sqrt[4]{3375 c^{3 \cdot 3}}}{15 c^{2}})}^{2}$

Step 3.3.2.1.11.4.2

Multiply $3$ by $3$ .

$f (- \frac{\sqrt[4]{15 c^{3}}}{c}) = \sqrt{15 c} - 5 {(- \frac{\sqrt[4]{3375 c^{9}}}{15 c^{2}})}^{2}$

Step 3.3.2.1.11.5

Rewrite $3375 c^{9}$ as ${(c^{2})}^{4} \cdot (3375 c)$ .

Tap for more steps...

Step 3.3.2.1.11.5.1

Factor out $c^{8}$ .

$f (- \frac{\sqrt[4]{15 c^{3}}}{c}) = \sqrt{15 c} - 5 {(- \frac{\sqrt[4]{3375 (c^{8} c)}}{15 c^{2}})}^{2}$

Step 3.3.2.1.11.5.2

Rewrite $c^{8}$ as ${(c^{2})}^{4}$ .

$f (- \frac{\sqrt[4]{15 c^{3}}}{c}) = \sqrt{15 c} - 5 {(- \frac{\sqrt[4]{3375 ({(c^{2})}^{4} c)}}{15 c^{2}})}^{2}$

Step 3.3.2.1.11.5.3

Reorder $3375$ and ${(c^{2})}^{4}$ .

$f (- \frac{\sqrt[4]{15 c^{3}}}{c}) = \sqrt{15 c} - 5 {(- \frac{\sqrt[4]{{(c^{2})}^{4} \cdot (3375 c)}}{15 c^{2}})}^{2}$

Step 3.3.2.1.11.5.4

Add parentheses.

$f (- \frac{\sqrt[4]{15 c^{3}}}{c}) = \sqrt{15 c} - 5 {(- \frac{\sqrt[4]{{(c^{2})}^{4} \cdot (3375 c)}}{15 c^{2}})}^{2}$

Step 3.3.2.1.11.6

Pull terms out from under the radical.

$f (- \frac{\sqrt[4]{15 c^{3}}}{c}) = \sqrt{15 c} - 5 {(- \frac{c^{2} \sqrt[4]{3375 c}}{15 c^{2}})}^{2}$

Step 3.3.2.1.12

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

Tap for more steps...

Step 3.3.2.1.12.1

Cancel the common factor.

$f (- \frac{\sqrt[4]{15 c^{3}}}{c}) = \sqrt{15 c} - 5 {(- \frac{c^{2} \sqrt[4]{3375 c}}{15 c^{2}})}^{2}$

Step 3.3.2.1.12.2

Rewrite the expression.

$f (- \frac{\sqrt[4]{15 c^{3}}}{c}) = \sqrt{15 c} - 5 {(- \frac{\sqrt[4]{3375 c}}{15})}^{2}$

Step 3.3.2.1.13

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

Tap for more steps...

Step 3.3.2.1.13.1

Apply the product rule to $- \frac{\sqrt[4]{3375 c}}{15}$ .

$f (- \frac{\sqrt[4]{15 c^{3}}}{c}) = \sqrt{15 c} - 5 ({(- 1)}^{2} {(\frac{\sqrt[4]{3375 c}}{15})}^{2})$

Step 3.3.2.1.13.2

Apply the product rule to $\frac{\sqrt[4]{3375 c}}{15}$ .

$f (- \frac{\sqrt[4]{15 c^{3}}}{c}) = \sqrt{15 c} - 5 ({(- 1)}^{2} (\frac{{\sqrt[4]{3375 c}}^{2}}{15^{2}}))$

Step 3.3.2.1.14

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

$f (- \frac{\sqrt[4]{15 c^{3}}}{c}) = \sqrt{15 c} - 5 (1 (\frac{{\sqrt[4]{3375 c}}^{2}}{15^{2}}))$

Step 3.3.2.1.15

Multiply $\frac{{\sqrt[4]{3375 c}}^{2}}{15^{2}}$ by $1$ .

$f (- \frac{\sqrt[4]{15 c^{3}}}{c}) = \sqrt{15 c} - 5 \frac{{\sqrt[4]{3375 c}}^{2}}{15^{2}}$

Step 3.3.2.1.16

Simplify the numerator.

Tap for more steps...

Step 3.3.2.1.16.1

Rewrite ${\sqrt[4]{3375 c}}^{2}$ as $\sqrt{3375 c}$ .

Tap for more steps...

Step 3.3.2.1.16.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{3375 c}$ as ${(3375 c)}^{\frac{1}{4}}$ .

$f (- \frac{\sqrt[4]{15 c^{3}}}{c}) = \sqrt{15 c} - 5 \frac{{({(3375 c)}^{\frac{1}{4}})}^{2}}{15^{2}}$

Step 3.3.2.1.16.1.2

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

$f (- \frac{\sqrt[4]{15 c^{3}}}{c}) = \sqrt{15 c} - 5 \frac{{(3375 c)}^{\frac{1}{4} \cdot 2}}{15^{2}}$

Step 3.3.2.1.16.1.3

Combine $\frac{1}{4}$ and $2$ .

$f (- \frac{\sqrt[4]{15 c^{3}}}{c}) = \sqrt{15 c} - 5 \frac{{(3375 c)}^{\frac{2}{4}}}{15^{2}}$

Step 3.3.2.1.16.1.4

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

Tap for more steps...

Step 3.3.2.1.16.1.4.1

Factor $2$ out of $2$ .

$f (- \frac{\sqrt[4]{15 c^{3}}}{c}) = \sqrt{15 c} - 5 \frac{{(3375 c)}^{\frac{2 (1)}{4}}}{15^{2}}$

Step 3.3.2.1.16.1.4.2

Cancel the common factors.

Tap for more steps...

Step 3.3.2.1.16.1.4.2.1

Factor $2$ out of $4$ .

$f (- \frac{\sqrt[4]{15 c^{3}}}{c}) = \sqrt{15 c} - 5 \frac{{(3375 c)}^{\frac{2 \cdot 1}{2 \cdot 2}}}{15^{2}}$

Step 3.3.2.1.16.1.4.2.2

Cancel the common factor.

$f (- \frac{\sqrt[4]{15 c^{3}}}{c}) = \sqrt{15 c} - 5 \frac{{(3375 c)}^{\frac{2 \cdot 1}{2 \cdot 2}}}{15^{2}}$

Step 3.3.2.1.16.1.4.2.3

Rewrite the expression.

$f (- \frac{\sqrt[4]{15 c^{3}}}{c}) = \sqrt{15 c} - 5 \frac{{(3375 c)}^{\frac{1}{2}}}{15^{2}}$

Step 3.3.2.1.16.1.5

Rewrite ${(3375 c)}^{\frac{1}{2}}$ as $\sqrt{3375 c}$ .

$f (- \frac{\sqrt[4]{15 c^{3}}}{c}) = \sqrt{15 c} - 5 \frac{\sqrt{3375 c}}{15^{2}}$

Step 3.3.2.1.16.2

Rewrite $3375 c$ as $15^{2} \cdot (15 c)$ .

Tap for more steps...

Step 3.3.2.1.16.2.1

Factor $225$ out of $3375$ .

$f (- \frac{\sqrt[4]{15 c^{3}}}{c}) = \sqrt{15 c} - 5 \frac{\sqrt{225 (15) c}}{15^{2}}$

Step 3.3.2.1.16.2.2

Rewrite $225$ as $15^{2}$ .

$f (- \frac{\sqrt[4]{15 c^{3}}}{c}) = \sqrt{15 c} - 5 \frac{\sqrt{15^{2} \cdot (15 c)}}{15^{2}}$

Step 3.3.2.1.16.2.3

Add parentheses.

$f (- \frac{\sqrt[4]{15 c^{3}}}{c}) = \sqrt{15 c} - 5 \frac{\sqrt{15^{2} \cdot (15 c)}}{15^{2}}$

Step 3.3.2.1.16.3

Pull terms out from under the radical.

$f (- \frac{\sqrt[4]{15 c^{3}}}{c}) = \sqrt{15 c} - 5 \frac{15 \sqrt{15 c}}{15^{2}}$

Step 3.3.2.1.17

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

$f (- \frac{\sqrt[4]{15 c^{3}}}{c}) = \sqrt{15 c} - 5 \frac{15 \sqrt{15 c}}{225}$

Step 3.3.2.1.18

Cancel the common factor of $5$ .

Tap for more steps...

Step 3.3.2.1.18.1

Factor $5$ out of $- 5$ .

$f (- \frac{\sqrt[4]{15 c^{3}}}{c}) = \sqrt{15 c} + 5 (- 1) (\frac{15 \sqrt{15 c}}{225})$

Step 3.3.2.1.18.2

Factor $5$ out of $225$ .

$f (- \frac{\sqrt[4]{15 c^{3}}}{c}) = \sqrt{15 c} + 5 \cdot (- 1 \frac{15 \sqrt{15 c}}{5 \cdot 45})$

Step 3.3.2.1.18.3

Cancel the common factor.

$f (- \frac{\sqrt[4]{15 c^{3}}}{c}) = \sqrt{15 c} + 5 \cdot (- 1 \frac{15 \sqrt{15 c}}{5 \cdot 45})$

Step 3.3.2.1.18.4

Rewrite the expression.

$f (- \frac{\sqrt[4]{15 c^{3}}}{c}) = \sqrt{15 c} - 1 \frac{15 \sqrt{15 c}}{45}$

Step 3.3.2.1.19

Cancel the common factor of $15$ and $45$ .

Tap for more steps...

Step 3.3.2.1.19.1

Factor $15$ out of $15 \sqrt{15 c}$ .

$f (- \frac{\sqrt[4]{15 c^{3}}}{c}) = \sqrt{15 c} - 1 \frac{15 (\sqrt{15 c})}{45}$

Step 3.3.2.1.19.2

Cancel the common factors.

Tap for more steps...

Step 3.3.2.1.19.2.1

Factor $15$ out of $45$ .

$f (- \frac{\sqrt[4]{15 c^{3}}}{c}) = \sqrt{15 c} - 1 \frac{15 \sqrt{15 c}}{15 \cdot 3}$

Step 3.3.2.1.19.2.2

Cancel the common factor.

$f (- \frac{\sqrt[4]{15 c^{3}}}{c}) = \sqrt{15 c} - 1 \frac{15 \sqrt{15 c}}{15 \cdot 3}$

Step 3.3.2.1.19.2.3

Rewrite the expression.

$f (- \frac{\sqrt[4]{15 c^{3}}}{c}) = \sqrt{15 c} - 1 \frac{\sqrt{15 c}}{3}$

Step 3.3.2.1.20

Rewrite $- 1 \frac{\sqrt{15 c}}{3}$ as $- \frac{\sqrt{15 c}}{3}$ .

$f (- \frac{\sqrt[4]{15 c^{3}}}{c}) = \sqrt{15 c} - \frac{\sqrt{15 c}}{3}$

Step 3.3.2.2

To write $\sqrt{15 c}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$f (- \frac{\sqrt[4]{15 c^{3}}}{c}) = \sqrt{15 c} \cdot \frac{3}{3} - \frac{\sqrt{15 c}}{3}$

Step 3.3.2.3

Simplify terms.

Tap for more steps...

Step 3.3.2.3.1

Combine $\sqrt{15 c}$ and $\frac{3}{3}$ .

$f (- \frac{\sqrt[4]{15 c^{3}}}{c}) = \frac{\sqrt{15 c} \cdot 3}{3} - \frac{\sqrt{15 c}}{3}$

Step 3.3.2.3.2

Combine the numerators over the common denominator.

$f (- \frac{\sqrt[4]{15 c^{3}}}{c}) = \frac{\sqrt{15 c} \cdot 3 - \sqrt{15 c}}{3}$

Step 3.3.2.4

Simplify the numerator.

Tap for more steps...

Step 3.3.2.4.1

Move $3$ to the left of $\sqrt{15 c}$ .

$f (- \frac{\sqrt[4]{15 c^{3}}}{c}) = \frac{3 \cdot \sqrt{15 c} - \sqrt{15 c}}{3}$

Step 3.3.2.4.2

Subtract $\sqrt{15 c}$ from $3 \sqrt{15 c}$ .

$f (- \frac{\sqrt[4]{15 c^{3}}}{c}) = \frac{2 \sqrt{15 c}}{3}$

Step 3.3.2.5

The final answer is $\frac{2 \sqrt{15 c}}{3}$ .

$\frac{2 \sqrt{15 c}}{3}$

Step 3.4

The point found by substituting $- \frac{\sqrt[4]{15 c^{3}}}{c}$ in $f (x) = c x^{2} - 5 x^{- 2}$ is $(- \frac{\sqrt[4]{15 c^{3}}}{c}, \frac{2 \sqrt{15 c}}{3})$ . This point can be an inflection point.

$(- \frac{\sqrt[4]{15 c^{3}}}{c}, \frac{2 \sqrt{15 c}}{3})$

Step 3.5

Determine the points that could be inflection points.

$(\frac{\sqrt[4]{15 c^{3}}}{c}, \frac{2 \sqrt{15 c}}{3}), (- \frac{\sqrt[4]{15 c^{3}}}{c}, \frac{2 \sqrt{15 c}}{3})$

Step 4

Split $(- \infty, \infty)$ into intervals around the points that could potentially be inflection points.

$(- \infty, \frac{\sqrt[4]{15 c^{3}}}{c}) \cup (\frac{\sqrt[4]{15 c^{3}}}{c}, \infty)$

Step 5

Substitute a value from the interval $(- \infty, \frac{\sqrt[4]{15 c^{3}}}{c})$ into the second derivative to determine if it is increasing or decreasing.

Tap for more steps...

Step 5.1

Replace the variable $x$ with $N^{2} a - 0.1$ in the expression.

$f'' (N^{2} a - 0.1) = 2 c - \frac{30}{{(N^{2} a - 0.1)}^{4}}$

Step 5.2

The final answer is $2 c - \frac{30}{{(N^{2} a - 0.1)}^{4}}$ .

$2 c - \frac{30}{{(N^{2} a - 0.1)}^{4}}$

Step 5.3

At $N^{2} a - 0.1$ , the second derivative is $N a N$ . Since this is negative, the second derivative is decreasing on the interval $(- \infty, \frac{\sqrt[4]{15 c^{3}}}{c})$

Decreasing on $(- \infty, \frac{\sqrt[4]{15 c^{3}}}{c})$ since $f'' (x) < 0$

Step 6

Substitute a value from the interval $(\frac{\sqrt[4]{15 c^{3}}}{c}, \infty)$ into the second derivative to determine if it is increasing or decreasing.

Tap for more steps...

Step 6.1

Replace the variable $x$ with $N^{2} a + 0.1$ in the expression.

$f'' (N^{2} a + 0.1) = 2 c - \frac{30}{{(N^{2} a + 0.1)}^{4}}$

Step 6.2

The final answer is $2 c - \frac{30}{{(N^{2} a + 0.1)}^{4}}$ .

$2 c - \frac{30}{{(N^{2} a + 0.1)}^{4}}$

Step 6.3

At $N^{2} a + 0.1$ , the second derivative is $N a N$ . Since this is negative, the second derivative is decreasing on the interval $(\frac{\sqrt[4]{15 c^{3}}}{c}, \infty)$

Decreasing on $(\frac{\sqrt[4]{15 c^{3}}}{c}, \infty)$ since $f'' (x) < 0$

Step 7

An inflection point is a point on a curve at which the concavity changes sign from plus to minus or from minus to plus. There are no points on the graph that satisfy these requirements.

No Inflection Points