Calculus Examples

$f (x) = 4 \sqrt{x} + 5$ , $[4, 7]$

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

By the Sum Rule, the derivative of $4 \sqrt{x} + 5$ with respect to $x$ is $\frac{d}{d x} [4 \sqrt{x}] + \frac{d}{d x} [5]$ .

$\frac{d}{d x} [4 \sqrt{x}] + \frac{d}{d x} [5]$

Step 1.1.1.2

Evaluate $\frac{d}{d x} [4 \sqrt{x}]$ .

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Step 1.1.1.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x}$ as $x^{\frac{1}{2}}$ .

$\frac{d}{d x} [4 x^{\frac{1}{2}}] + \frac{d}{d x} [5]$

Step 1.1.1.2.2

Since $4$ is constant with respect to $x$ , the derivative of $4 x^{\frac{1}{2}}$ with respect to $x$ is $4 \frac{d}{d x} [x^{\frac{1}{2}}]$ .

$4 \frac{d}{d x} [x^{\frac{1}{2}}] + \frac{d}{d x} [5]$

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

$4 (\frac{1}{2} x^{\frac{1}{2} - 1}) + \frac{d}{d x} [5]$

Step 1.1.1.2.4

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

$4 (\frac{1}{2} x^{\frac{1}{2} - 1 \cdot \frac{2}{2}}) + \frac{d}{d x} [5]$

Step 1.1.1.2.5

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

$4 (\frac{1}{2} x^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}}) + \frac{d}{d x} [5]$

Step 1.1.1.2.6

Combine the numerators over the common denominator.

$4 (\frac{1}{2} x^{\frac{1 - 1 \cdot 2}{2}}) + \frac{d}{d x} [5]$

Step 1.1.1.2.7

Simplify the numerator.

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Step 1.1.1.2.7.1

Multiply $- 1$ by $2$ .

$4 (\frac{1}{2} x^{\frac{1 - 2}{2}}) + \frac{d}{d x} [5]$

Step 1.1.1.2.7.2

Subtract $2$ from $1$ .

$4 (\frac{1}{2} x^{\frac{- 1}{2}}) + \frac{d}{d x} [5]$

Step 1.1.1.2.8

Move the negative in front of the fraction.

$4 (\frac{1}{2} x^{- \frac{1}{2}}) + \frac{d}{d x} [5]$

Step 1.1.1.2.9

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

$4 \frac{x^{- \frac{1}{2}}}{2} + \frac{d}{d x} [5]$

Step 1.1.1.2.10

Combine $4$ and $\frac{x^{- \frac{1}{2}}}{2}$ .

$\frac{4 x^{- \frac{1}{2}}}{2} + \frac{d}{d x} [5]$

Step 1.1.1.2.11

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

$\frac{4}{2 x^{\frac{1}{2}}} + \frac{d}{d x} [5]$

Step 1.1.1.2.12

Factor $2$ out of $4$ .

$\frac{2 \cdot 2}{2 x^{\frac{1}{2}}} + \frac{d}{d x} [5]$

Step 1.1.1.2.13

Cancel the common factors.

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Step 1.1.1.2.13.1

Factor $2$ out of $2 x^{\frac{1}{2}}$ .

$\frac{2 \cdot 2}{2 (x^{\frac{1}{2}})} + \frac{d}{d x} [5]$

Step 1.1.1.2.13.2

Cancel the common factor.

$\frac{2 \cdot 2}{2 x^{\frac{1}{2}}} + \frac{d}{d x} [5]$

Step 1.1.1.2.13.3

Rewrite the expression.

$\frac{2}{x^{\frac{1}{2}}} + \frac{d}{d x} [5]$

Step 1.1.1.3

Differentiate using the Constant Rule.

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Step 1.1.1.3.1

Since $5$ is constant with respect to $x$ , the derivative of $5$ with respect to $x$ is $0$ .

$\frac{2}{x^{\frac{1}{2}}} + 0$

Step 1.1.1.3.2

Add $\frac{2}{x^{\frac{1}{2}}}$ and $0$ .

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

Step 1.1.2

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

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

Step 1.2

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

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Step 1.2.1

Set the first derivative equal to $0$ .

$\frac{2}{x^{\frac{1}{2}}} = 0$

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 1.3

Find the values where the derivative is undefined.

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Step 1.3.1

Convert expressions with fractional exponents to radicals.

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Step 1.3.1.1

Apply the rule $x^{\frac{m}{n}} = \sqrt[n]{x^{m}}$ to rewrite the exponentiation as a radical.

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

Step 1.3.1.2

Anything raised to $1$ is the base itself.

$\frac{2}{\sqrt{x}}$

Step 1.3.2

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

$\sqrt{x} = 0$

Step 1.3.3

Solve for $x$ .

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Step 1.3.3.1

To remove the radical on the left side of the equation, square both sides of the equation.

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

Step 1.3.3.2

Simplify each side of the equation.

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Step 1.3.3.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x}$ as $x^{\frac{1}{2}}$ .

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

Step 1.3.3.2.2

Simplify the left side.

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Step 1.3.3.2.2.1

Simplify ${(x^{\frac{1}{2}})}^{2}$ .

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Step 1.3.3.2.2.1.1

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

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Step 1.3.3.2.2.1.1.1

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

$x^{\frac{1}{2} \cdot 2} = 0^{2}$

Step 1.3.3.2.2.1.1.2

Cancel the common factor of $2$ .

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Step 1.3.3.2.2.1.1.2.1

Cancel the common factor.

$x^{\frac{1}{2} \cdot 2} = 0^{2}$

Step 1.3.3.2.2.1.1.2.2

Rewrite the expression.

$x^{1} = 0^{2}$

Step 1.3.3.2.2.1.2

Simplify.

$x = 0^{2}$

Step 1.3.3.2.3

Simplify the right side.

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Step 1.3.3.2.3.1

Raising $0$ to any positive power yields $0$ .

$x = 0$

Step 1.3.4

Set the radicand in $\sqrt{x}$ less than $0$ to find where the expression is undefined.

$x < 0$

Step 1.3.5

The equation is undefined where the denominator equals $0$ , the argument of a square root is less than $0$ , or the argument of a logarithm is less than or equal to $0$ .

$x \leq 0$

$(- \infty, 0]$

$x \leq 0$

$(- \infty, 0]$

Step 1.4

Evaluate $4 \sqrt{x} + 5$ at each $x$ value where the derivative is $0$ or undefined.

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Step 1.4.1

Evaluate at $x = 0$ .

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Step 1.4.1.1

Substitute $0$ for $x$ .

$4 \sqrt{0} + 5$

Step 1.4.1.2

Simplify.

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Step 1.4.1.2.1

Remove parentheses.

$4 \sqrt{0} + 5$

Step 1.4.1.2.2

Simplify each term.

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Step 1.4.1.2.2.1

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

$4 \sqrt{0^{2}} + 5$

Step 1.4.1.2.2.2

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

$4 \cdot 0 + 5$

Step 1.4.1.2.2.3

Multiply $4$ by $0$ .

$0 + 5$

Step 1.4.1.2.3

Add $0$ and $5$ .

$5$

Step 1.4.2

List all of the points.

$(0, 5)$

Step 2

Exclude the points that are not on the interval.

Step 3

Evaluate at the included endpoints.

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Step 3.1

Evaluate at $x = 4$ .

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Step 3.1.1

Substitute $4$ for $x$ .

$4 \sqrt{4} + 5$

Step 3.1.2

Simplify.

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Step 3.1.2.1

Remove parentheses.

$4 \sqrt{4} + 5$

Step 3.1.2.2

Simplify each term.

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Step 3.1.2.2.1

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

$4 \sqrt{2^{2}} + 5$

Step 3.1.2.2.2

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

$4 \cdot 2 + 5$

Step 3.1.2.2.3

Multiply $4$ by $2$ .

$8 + 5$

Step 3.1.2.3

Add $8$ and $5$ .

$13$

Step 3.2

Evaluate at $x = 7$ .

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Step 3.2.1

Substitute $7$ for $x$ .

$4 \sqrt{7} + 5$

Step 3.2.2

Remove parentheses.

$4 \sqrt{7} + 5$

Step 3.3

List all of the points.

$(4, 13), (7, 4 \sqrt{7} + 5)$

Step 4

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.

Absolute Maximum: $(7, 4 \sqrt{7} + 5)$

Absolute Minimum: $(4, 13)$

Step 5