Calculus Examples

$f (x) = \sqrt{x}$

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 $\sqrt{x}$ as $x^{\frac{1}{2}}$ .

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

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 $n = \frac{1}{2}$ .

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

Step 1.1.1.3

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 1.1.1.4

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

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

Step 1.1.1.5

Combine the numerators over the common denominator.

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

Step 1.1.1.6

Simplify the numerator.

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Step 1.1.1.6.1

Multiply $- 1$ by $2$ .

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

Step 1.1.1.6.2

Subtract $2$ from $1$ .

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

Step 1.1.1.7

Move the negative in front of the fraction.

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

Step 1.1.1.8

Simplify.

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Step 1.1.1.8.1

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

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

Step 1.1.1.8.2

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

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

Step 1.1.2

Find the second derivative.

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Step 1.1.2.1

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

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

Step 1.1.2.2

Apply basic rules of exponents.

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Step 1.1.2.2.1

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

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

Step 1.1.2.2.2

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

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Step 1.1.2.2.2.1

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

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

Step 1.1.2.2.2.2

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

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

Step 1.1.2.2.2.3

Move the negative in front of the fraction.

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

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 $n = - \frac{1}{2}$ .

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

Step 1.1.2.4

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 1.1.2.5

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

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

Step 1.1.2.6

Combine the numerators over the common denominator.

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

Step 1.1.2.7

Simplify the numerator.

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Step 1.1.2.7.1

Multiply $- 1$ by $2$ .

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

Step 1.1.2.7.2

Subtract $2$ from $- 1$ .

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

Step 1.1.2.8

Move the negative in front of the fraction.

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

Step 1.1.2.9

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

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

Step 1.1.2.10

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

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

Step 1.1.2.11

Simplify the expression.

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Step 1.1.2.11.1

Multiply $2$ by $2$ .

$- \frac{x^{- \frac{3}{2}}}{4}$

Step 1.1.2.11.2

Move $x^{- \frac{3}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$f'' (x) = - \frac{1}{4 x^{\frac{3}{2}}}$

Step 1.1.3

The second derivative of $f (x)$ with respect to $x$ is $- \frac{1}{4 x^{\frac{3}{2}}}$ .

$- \frac{1}{4 x^{\frac{3}{2}}}$

Step 1.2

Set the second derivative equal to $0$ then solve the equation $- \frac{1}{4 x^{\frac{3}{2}}} = 0$ .

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Step 1.2.1

Set the second derivative equal to $0$ .

$- \frac{1}{4 x^{\frac{3}{2}}} = 0$

Step 1.2.2

Set the numerator equal to zero.

$1 = 0$

Step 1.2.3

Since $1 \neq 0$ , there are no solutions.

No solution

Step 2

Find the domain of $f (x) = \sqrt{x}$ .

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Step 2.1

Set the radicand in $\sqrt{x}$ greater than or equal to $0$ to find where the expression is defined.

$x \geq 0$

Step 2.2

The domain is all values of $x$ that make the expression defined.

Interval Notation:

$[0, \infty)$

Set-Builder Notation:

${x | x \geq 0}$

Interval Notation:

$[0, \infty)$

Set-Builder Notation:

${x | x \geq 0}$

Step 3

Create intervals around the $x$ -values where the second derivative is zero or undefined.

$[0, \infty)$

Step 4

Substitute any number from the interval $[0, \infty)$ into the second derivative and evaluate to determine the concavity.

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Step 4.1

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

$f'' (2) = - \frac{1}{4 {(2)}^{\frac{3}{2}}}$

Step 4.2

Simplify the result.

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Step 4.2.1

Simplify the denominator.

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Step 4.2.1.1

Rewrite $4$ as $2^{2}$ .

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

Step 4.2.1.2

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

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

Step 4.2.1.3

To write $2$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$f'' (2) = - \frac{1}{2^{2 \cdot \frac{2}{2} + \frac{3}{2}}}$

Step 4.2.1.4

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

$f'' (2) = - \frac{1}{2^{\frac{2 \cdot 2}{2} + \frac{3}{2}}}$

Step 4.2.1.5

Combine the numerators over the common denominator.

$f'' (2) = - \frac{1}{2^{\frac{2 \cdot 2 + 3}{2}}}$

Step 4.2.1.6

Simplify the numerator.

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Step 4.2.1.6.1

Multiply $2$ by $2$ .

$f'' (2) = - \frac{1}{2^{\frac{4 + 3}{2}}}$

Step 4.2.1.6.2

Add $4$ and $3$ .

$f'' (2) = - \frac{1}{2^{\frac{7}{2}}}$

Step 4.2.2

The final answer is $- \frac{1}{2^{\frac{7}{2}}}$ .

$- \frac{1}{2^{\frac{7}{2}}}$

Step 4.3

The graph is concave down on the interval $[0, \infty)$ because $f'' (2)$ is negative.

Concave down on $[0, \infty)$ since $f'' (x)$ is negative

Step 5