Calculus Examples

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

Step 1

Find the $x$ values where the second derivative is equal to $0$ .

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Step 1.1

Find the second derivative.

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Step 1.1.1

Find the first derivative.

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Step 1.1.1.1

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

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

Step 1.1.1.2

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

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

Step 1.1.1.3

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{1}{3} x^{\frac{1}{3} - 1 \cdot \frac{3}{3}}$

Step 1.1.1.4

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

$\frac{1}{3} x^{\frac{1}{3} + \frac{- 1 \cdot 3}{3}}$

Step 1.1.1.5

Combine the numerators over the common denominator.

$\frac{1}{3} x^{\frac{1 - 1 \cdot 3}{3}}$

Step 1.1.1.6

Simplify the numerator.

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Step 1.1.1.6.1

Multiply $- 1$ by $3$ .

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

Step 1.1.1.6.2

Subtract $3$ from $1$ .

$\frac{1}{3} x^{\frac{- 2}{3}}$

Step 1.1.1.7

Move the negative in front of the fraction.

$\frac{1}{3} x^{- \frac{2}{3}}$

Step 1.1.1.8

Simplify.

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Step 1.1.1.8.1

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

$\frac{1}{3} \cdot \frac{1}{x^{\frac{2}{3}}}$

Step 1.1.1.8.2

Multiply $\frac{1}{3}$ by $\frac{1}{x^{\frac{2}{3}}}$ .

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

Step 1.1.2

Find the second derivative.

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Step 1.1.2.1

Since $\frac{1}{3}$ is constant with respect to $x$ , the derivative of $\frac{1}{3 x^{\frac{2}{3}}}$ with respect to $x$ is $\frac{1}{3} \frac{d}{d x} [\frac{1}{x^{\frac{2}{3}}}]$ .

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

Step 1.1.2.2

Apply basic rules of exponents.

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Step 1.1.2.2.1

Rewrite $\frac{1}{x^{\frac{2}{3}}}$ as ${(x^{\frac{2}{3}})}^{- 1}$ .

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

Step 1.1.2.2.2

Multiply the exponents in ${(x^{\frac{2}{3}})}^{- 1}$ .

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Step 1.1.2.2.2.1

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

$\frac{1}{3} \frac{d}{d x} [x^{\frac{2}{3} \cdot - 1}]$

Step 1.1.2.2.2.2

Multiply $\frac{2}{3} \cdot - 1$ .

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Step 1.1.2.2.2.2.1

Combine $\frac{2}{3}$ and $- 1$ .

$\frac{1}{3} \frac{d}{d x} [x^{\frac{2 \cdot - 1}{3}}]$

Step 1.1.2.2.2.2.2

Multiply $2$ by $- 1$ .

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

Step 1.1.2.2.2.3

Move the negative in front of the fraction.

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

Step 1.1.2.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = - \frac{2}{3}$ .

$\frac{1}{3} (- \frac{2}{3} x^{- \frac{2}{3} - 1})$

Step 1.1.2.4

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{1}{3} (- \frac{2}{3} x^{- \frac{2}{3} - 1 \cdot \frac{3}{3}})$

Step 1.1.2.5

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

$\frac{1}{3} (- \frac{2}{3} x^{- \frac{2}{3} + \frac{- 1 \cdot 3}{3}})$

Step 1.1.2.6

Combine the numerators over the common denominator.

$\frac{1}{3} (- \frac{2}{3} x^{\frac{- 2 - 1 \cdot 3}{3}})$

Step 1.1.2.7

Simplify the numerator.

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Step 1.1.2.7.1

Multiply $- 1$ by $3$ .

$\frac{1}{3} (- \frac{2}{3} x^{\frac{- 2 - 3}{3}})$

Step 1.1.2.7.2

Subtract $3$ from $- 2$ .

$\frac{1}{3} (- \frac{2}{3} x^{\frac{- 5}{3}})$

Step 1.1.2.8

Move the negative in front of the fraction.

$\frac{1}{3} (- \frac{2}{3} x^{- \frac{5}{3}})$

Step 1.1.2.9

Combine $x^{- \frac{5}{3}}$ and $\frac{2}{3}$ .

$\frac{1}{3} (- \frac{x^{- \frac{5}{3}} \cdot 2}{3})$

Step 1.1.2.10

Multiply $\frac{1}{3}$ by $\frac{x^{- \frac{5}{3}} \cdot 2}{3}$ .

$- \frac{x^{- \frac{5}{3}} \cdot 2}{3 \cdot 3}$

Step 1.1.2.11

Simplify the expression.

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Step 1.1.2.11.1

Multiply $3$ by $3$ .

$- \frac{x^{- \frac{5}{3}} \cdot 2}{9}$

Step 1.1.2.11.2

Move $2$ to the left of $x^{- \frac{5}{3}}$ .

$- \frac{2 \cdot x^{- \frac{5}{3}}}{9}$

Step 1.1.2.11.3

Move $x^{- \frac{5}{3}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$f'' (x) = - \frac{2}{9 x^{\frac{5}{3}}}$

Step 1.1.3

The second derivative of $p (x)$ with respect to $x$ is $- \frac{2}{9 x^{\frac{5}{3}}}$ .

$- \frac{2}{9 x^{\frac{5}{3}}}$

Step 1.2

Set the second derivative equal to $0$ then solve the equation $- \frac{2}{9 x^{\frac{5}{3}}} = 0$ .

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Step 1.2.1

Set the second derivative equal to $0$ .

$- \frac{2}{9 x^{\frac{5}{3}}} = 0$

Step 1.2.2

Set the numerator equal to zero.

$2 = 0$

Step 1.2.3

Since $2 \neq 0$ , there are no solutions.

No solution

Step 2

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.

Interval Notation:

$(- \infty, \infty)$

Set-Builder Notation:

${x | x \in ℝ}$

Step 3

The graph is concave down because the second derivative is negative.

The graph is concave down

Step 4