Calculus Examples

$f (x) = \frac{1}{3} x^{3} - 4 x^{2} + 13 x$ ; $[1, 6]$

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 $\frac{1}{3} x^{3} - 4 x^{2} + 13 x$ with respect to $x$ is $\frac{d}{d x} [\frac{1}{3} x^{3}] + \frac{d}{d x} [- 4 x^{2}] + \frac{d}{d x} [13 x]$ .

$f' (x) = \frac{d}{d x} (\frac{1}{3} x^{3}) + \frac{d}{d x} (- 4 x^{2}) + \frac{d}{d x} (13 x)$

Step 1.1.1.2

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

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Step 1.1.1.2.1

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

$f' (x) = \frac{1}{3} \cdot \frac{d}{d x} (x^{3}) + \frac{d}{d x} (- 4 x^{2}) + \frac{d}{d x} (13 x)$

Step 1.1.1.2.2

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

$f' (x) = \frac{1}{3} \cdot (3 x^{2}) + \frac{d}{d x} (- 4 x^{2}) + \frac{d}{d x} (13 x)$

Step 1.1.1.2.3

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

$f' (x) = \frac{3}{3} x^{2} + \frac{d}{d x} (- 4 x^{2}) + \frac{d}{d x} (13 x)$

Step 1.1.1.2.4

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

$f' (x) = \frac{3 x^{2}}{3} + \frac{d}{d x} (- 4 x^{2}) + \frac{d}{d x} (13 x)$

Step 1.1.1.2.5

Cancel the common factor of $3$ .

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Step 1.1.1.2.5.1

Cancel the common factor.

$f' (x) = \frac{3 x^{2}}{3} + \frac{d}{d x} (- 4 x^{2}) + \frac{d}{d x} (13 x)$

Step 1.1.1.2.5.2

Divide $x^{2}$ by $1$ .

$f' (x) = x^{2} + \frac{d}{d x} (- 4 x^{2}) + \frac{d}{d x} (13 x)$

Step 1.1.1.3

Evaluate $\frac{d}{d x} [- 4 x^{2}]$ .

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Step 1.1.1.3.1

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

$f' (x) = x^{2} - 4 \frac{d}{d x} x^{2} + \frac{d}{d x} (13 x)$

Step 1.1.1.3.2

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

$f' (x) = x^{2} - 4 (2 x) + \frac{d}{d x} (13 x)$

Step 1.1.1.3.3

Multiply $2$ by $- 4$ .

$f' (x) = x^{2} - 8 x + \frac{d}{d x} (13 x)$

Step 1.1.1.4

Evaluate $\frac{d}{d x} [13 x]$ .

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Step 1.1.1.4.1

Since $13$ is constant with respect to $x$ , the derivative of $13 x$ with respect to $x$ is $13 \frac{d}{d x} [x]$ .

$f' (x) = x^{2} - 8 x + 13 \frac{d}{d x} (x)$

Step 1.1.1.4.2

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

$f' (x) = x^{2} - 8 x + 13 \cdot 1$

Step 1.1.1.4.3

Multiply $13$ by $1$ .

$f' (x) = x^{2} - 8 x + 13$

Step 1.1.2

The first derivative of $f (x)$ with respect to $x$ is $x^{2} - 8 x + 13$ .

$x^{2} - 8 x + 13$

Step 1.2

Set the first derivative equal to $0$ then solve the equation $x^{2} - 8 x + 13 = 0$ .

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Step 1.2.1

Set the first derivative equal to $0$ .

$x^{2} - 8 x + 13 = 0$

Step 1.2.2

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 1.2.3

Substitute the values $a = 1$ , $b = - 8$ , and $c = 13$ into the quadratic formula and solve for $x$ .

$\frac{8 \pm \sqrt{{(- 8)}^{2} - 4 \cdot (1 \cdot 13)}}{2 \cdot 1}$

Step 1.2.4

Simplify.

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Step 1.2.4.1

Simplify the numerator.

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Step 1.2.4.1.1

Raise $- 8$ to the power of $2$ .

$x = \frac{8 \pm \sqrt{64 - 4 \cdot 1 \cdot 13}}{2 \cdot 1}$

Step 1.2.4.1.2

Multiply $- 4 \cdot 1 \cdot 13$ .

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Step 1.2.4.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{8 \pm \sqrt{64 - 4 \cdot 13}}{2 \cdot 1}$

Step 1.2.4.1.2.2

Multiply $- 4$ by $13$ .

$x = \frac{8 \pm \sqrt{64 - 52}}{2 \cdot 1}$

Step 1.2.4.1.3

Subtract $52$ from $64$ .

$x = \frac{8 \pm \sqrt{12}}{2 \cdot 1}$

Step 1.2.4.1.4

Rewrite $12$ as $2^{2} \cdot 3$ .

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Step 1.2.4.1.4.1

Factor $4$ out of $12$ .

$x = \frac{8 \pm \sqrt{4 (3)}}{2 \cdot 1}$

Step 1.2.4.1.4.2

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

$x = \frac{8 \pm \sqrt{2^{2} \cdot 3}}{2 \cdot 1}$

Step 1.2.4.1.5

Pull terms out from under the radical.

$x = \frac{8 \pm 2 \sqrt{3}}{2 \cdot 1}$

Step 1.2.4.2

Multiply $2$ by $1$ .

$x = \frac{8 \pm 2 \sqrt{3}}{2}$

Step 1.2.4.3

Simplify $\frac{8 \pm 2 \sqrt{3}}{2}$ .

$x = 4 \pm \sqrt{3}$

Step 1.2.5

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Step 1.2.5.1

Simplify the numerator.

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Step 1.2.5.1.1

Raise $- 8$ to the power of $2$ .

$x = \frac{8 \pm \sqrt{64 - 4 \cdot 1 \cdot 13}}{2 \cdot 1}$

Step 1.2.5.1.2

Multiply $- 4 \cdot 1 \cdot 13$ .

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Step 1.2.5.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{8 \pm \sqrt{64 - 4 \cdot 13}}{2 \cdot 1}$

Step 1.2.5.1.2.2

Multiply $- 4$ by $13$ .

$x = \frac{8 \pm \sqrt{64 - 52}}{2 \cdot 1}$

Step 1.2.5.1.3

Subtract $52$ from $64$ .

$x = \frac{8 \pm \sqrt{12}}{2 \cdot 1}$

Step 1.2.5.1.4

Rewrite $12$ as $2^{2} \cdot 3$ .

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Step 1.2.5.1.4.1

Factor $4$ out of $12$ .

$x = \frac{8 \pm \sqrt{4 (3)}}{2 \cdot 1}$

Step 1.2.5.1.4.2

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

$x = \frac{8 \pm \sqrt{2^{2} \cdot 3}}{2 \cdot 1}$

Step 1.2.5.1.5

Pull terms out from under the radical.

$x = \frac{8 \pm 2 \sqrt{3}}{2 \cdot 1}$

Step 1.2.5.2

Multiply $2$ by $1$ .

$x = \frac{8 \pm 2 \sqrt{3}}{2}$

Step 1.2.5.3

Simplify $\frac{8 \pm 2 \sqrt{3}}{2}$ .

$x = 4 \pm \sqrt{3}$

Step 1.2.5.4

Change the $\pm$ to $+$ .

$x = 4 + \sqrt{3}$

Step 1.2.6

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Step 1.2.6.1

Simplify the numerator.

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Step 1.2.6.1.1

Raise $- 8$ to the power of $2$ .

$x = \frac{8 \pm \sqrt{64 - 4 \cdot 1 \cdot 13}}{2 \cdot 1}$

Step 1.2.6.1.2

Multiply $- 4 \cdot 1 \cdot 13$ .

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Step 1.2.6.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{8 \pm \sqrt{64 - 4 \cdot 13}}{2 \cdot 1}$

Step 1.2.6.1.2.2

Multiply $- 4$ by $13$ .

$x = \frac{8 \pm \sqrt{64 - 52}}{2 \cdot 1}$

Step 1.2.6.1.3

Subtract $52$ from $64$ .

$x = \frac{8 \pm \sqrt{12}}{2 \cdot 1}$

Step 1.2.6.1.4

Rewrite $12$ as $2^{2} \cdot 3$ .

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Step 1.2.6.1.4.1

Factor $4$ out of $12$ .

$x = \frac{8 \pm \sqrt{4 (3)}}{2 \cdot 1}$

Step 1.2.6.1.4.2

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

$x = \frac{8 \pm \sqrt{2^{2} \cdot 3}}{2 \cdot 1}$

Step 1.2.6.1.5

Pull terms out from under the radical.

$x = \frac{8 \pm 2 \sqrt{3}}{2 \cdot 1}$

Step 1.2.6.2

Multiply $2$ by $1$ .

$x = \frac{8 \pm 2 \sqrt{3}}{2}$

Step 1.2.6.3

Simplify $\frac{8 \pm 2 \sqrt{3}}{2}$ .

$x = 4 \pm \sqrt{3}$

Step 1.2.6.4

Change the $\pm$ to $-$ .

$x = 4 - \sqrt{3}$

Step 1.2.7

The final answer is the combination of both solutions.

$x = 4 + \sqrt{3}, 4 - \sqrt{3}$

Step 1.3

Find the values where the derivative is undefined.

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Step 1.3.1

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.

Step 1.4

Evaluate $\frac{1}{3} x^{3} - 4 x^{2} + 13 x$ at each $x$ value where the derivative is $0$ or undefined.

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Step 1.4.1

Evaluate at $x = 4 + \sqrt{3}$ .

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Step 1.4.1.1

Substitute $4 + \sqrt{3}$ for $x$ .

$\frac{1}{3} \cdot {(4 + \sqrt{3})}^{3} - 4 {(4 + \sqrt{3})}^{2} + 13 (4 + \sqrt{3})$

Step 1.4.1.2

Simplify.

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Step 1.4.1.2.1

Simplify each term.

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Step 1.4.1.2.1.1

Use the Binomial Theorem.

$\frac{1}{3} \cdot (4^{3} + 3 \cdot 4^{2} \sqrt{3} + 3 \cdot 4 {\sqrt{3}}^{2} + {\sqrt{3}}^{3}) - 4 {(4 + \sqrt{3})}^{2} + 13 (4 + \sqrt{3})$

Step 1.4.1.2.1.2

Simplify each term.

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Step 1.4.1.2.1.2.1

Raise $4$ to the power of $3$ .

$\frac{1}{3} \cdot (64 + 3 \cdot 4^{2} \sqrt{3} + 3 \cdot 4 {\sqrt{3}}^{2} + {\sqrt{3}}^{3}) - 4 {(4 + \sqrt{3})}^{2} + 13 (4 + \sqrt{3})$

Step 1.4.1.2.1.2.2

Raise $4$ to the power of $2$ .

$\frac{1}{3} \cdot (64 + 3 \cdot 16 \sqrt{3} + 3 \cdot 4 {\sqrt{3}}^{2} + {\sqrt{3}}^{3}) - 4 {(4 + \sqrt{3})}^{2} + 13 (4 + \sqrt{3})$

Step 1.4.1.2.1.2.3

Multiply $3$ by $16$ .

$\frac{1}{3} \cdot (64 + 48 \sqrt{3} + 3 \cdot 4 {\sqrt{3}}^{2} + {\sqrt{3}}^{3}) - 4 {(4 + \sqrt{3})}^{2} + 13 (4 + \sqrt{3})$

Step 1.4.1.2.1.2.4

Multiply $3$ by $4$ .

$\frac{1}{3} \cdot (64 + 48 \sqrt{3} + 12 {\sqrt{3}}^{2} + {\sqrt{3}}^{3}) - 4 {(4 + \sqrt{3})}^{2} + 13 (4 + \sqrt{3})$

Step 1.4.1.2.1.2.5

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 1.4.1.2.1.2.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$\frac{1}{3} \cdot (64 + 48 \sqrt{3} + 12 {(3^{\frac{1}{2}})}^{2} + {\sqrt{3}}^{3}) - 4 {(4 + \sqrt{3})}^{2} + 13 (4 + \sqrt{3})$

Step 1.4.1.2.1.2.5.2

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

$\frac{1}{3} \cdot (64 + 48 \sqrt{3} + 12 \cdot 3^{\frac{1}{2} \cdot 2} + {\sqrt{3}}^{3}) - 4 {(4 + \sqrt{3})}^{2} + 13 (4 + \sqrt{3})$

Step 1.4.1.2.1.2.5.3

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

$\frac{1}{3} \cdot (64 + 48 \sqrt{3} + 12 \cdot 3^{\frac{2}{2}} + {\sqrt{3}}^{3}) - 4 {(4 + \sqrt{3})}^{2} + 13 (4 + \sqrt{3})$

Step 1.4.1.2.1.2.5.4

Cancel the common factor of $2$ .

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Step 1.4.1.2.1.2.5.4.1

Cancel the common factor.

$\frac{1}{3} \cdot (64 + 48 \sqrt{3} + 12 \cdot 3^{\frac{2}{2}} + {\sqrt{3}}^{3}) - 4 {(4 + \sqrt{3})}^{2} + 13 (4 + \sqrt{3})$

Step 1.4.1.2.1.2.5.4.2

Rewrite the expression.

$\frac{1}{3} \cdot (64 + 48 \sqrt{3} + 12 \cdot 3^{1} + {\sqrt{3}}^{3}) - 4 {(4 + \sqrt{3})}^{2} + 13 (4 + \sqrt{3})$

Step 1.4.1.2.1.2.5.5

Evaluate the exponent.

$\frac{1}{3} \cdot (64 + 48 \sqrt{3} + 12 \cdot 3 + {\sqrt{3}}^{3}) - 4 {(4 + \sqrt{3})}^{2} + 13 (4 + \sqrt{3})$

Step 1.4.1.2.1.2.6

Multiply $12$ by $3$ .

$\frac{1}{3} \cdot (64 + 48 \sqrt{3} + 36 + {\sqrt{3}}^{3}) - 4 {(4 + \sqrt{3})}^{2} + 13 (4 + \sqrt{3})$

Step 1.4.1.2.1.2.7

Rewrite ${\sqrt{3}}^{3}$ as $\sqrt{3^{3}}$ .

$\frac{1}{3} \cdot (64 + 48 \sqrt{3} + 36 + \sqrt{3^{3}}) - 4 {(4 + \sqrt{3})}^{2} + 13 (4 + \sqrt{3})$

Step 1.4.1.2.1.2.8

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

$\frac{1}{3} \cdot (64 + 48 \sqrt{3} + 36 + \sqrt{27}) - 4 {(4 + \sqrt{3})}^{2} + 13 (4 + \sqrt{3})$

Step 1.4.1.2.1.2.9

Rewrite $27$ as $3^{2} \cdot 3$ .

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Step 1.4.1.2.1.2.9.1

Factor $9$ out of $27$ .

$\frac{1}{3} \cdot (64 + 48 \sqrt{3} + 36 + \sqrt{9 (3)}) - 4 {(4 + \sqrt{3})}^{2} + 13 (4 + \sqrt{3})$

Step 1.4.1.2.1.2.9.2

Rewrite $9$ as $3^{2}$ .

$\frac{1}{3} \cdot (64 + 48 \sqrt{3} + 36 + \sqrt{3^{2} \cdot 3}) - 4 {(4 + \sqrt{3})}^{2} + 13 (4 + \sqrt{3})$

Step 1.4.1.2.1.2.10

Pull terms out from under the radical.

$\frac{1}{3} \cdot (64 + 48 \sqrt{3} + 36 + 3 \sqrt{3}) - 4 {(4 + \sqrt{3})}^{2} + 13 (4 + \sqrt{3})$

Step 1.4.1.2.1.3

Add $64$ and $36$ .

$\frac{1}{3} \cdot (100 + 48 \sqrt{3} + 3 \sqrt{3}) - 4 {(4 + \sqrt{3})}^{2} + 13 (4 + \sqrt{3})$

Step 1.4.1.2.1.4

Add $48 \sqrt{3}$ and $3 \sqrt{3}$ .

$\frac{1}{3} \cdot (100 + 51 \sqrt{3}) - 4 {(4 + \sqrt{3})}^{2} + 13 (4 + \sqrt{3})$

Step 1.4.1.2.1.5

Apply the distributive property.

$\frac{1}{3} \cdot 100 + \frac{1}{3} (51 \sqrt{3}) - 4 {(4 + \sqrt{3})}^{2} + 13 (4 + \sqrt{3})$

Step 1.4.1.2.1.6

Combine $\frac{1}{3}$ and $100$ .

$\frac{100}{3} + \frac{1}{3} (51 \sqrt{3}) - 4 {(4 + \sqrt{3})}^{2} + 13 (4 + \sqrt{3})$

Step 1.4.1.2.1.7

Cancel the common factor of $3$ .

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Step 1.4.1.2.1.7.1

Factor $3$ out of $51 \sqrt{3}$ .

$\frac{100}{3} + \frac{1}{3} (3 (17 \sqrt{3})) - 4 {(4 + \sqrt{3})}^{2} + 13 (4 + \sqrt{3})$

Step 1.4.1.2.1.7.2

Cancel the common factor.

$\frac{100}{3} + \frac{1}{3} (3 (17 \sqrt{3})) - 4 {(4 + \sqrt{3})}^{2} + 13 (4 + \sqrt{3})$

Step 1.4.1.2.1.7.3

Rewrite the expression.

$\frac{100}{3} + 17 \sqrt{3} - 4 {(4 + \sqrt{3})}^{2} + 13 (4 + \sqrt{3})$

Step 1.4.1.2.1.8

Rewrite ${(4 + \sqrt{3})}^{2}$ as $(4 + \sqrt{3}) (4 + \sqrt{3})$ .

$\frac{100}{3} + 17 \sqrt{3} - 4 ((4 + \sqrt{3}) (4 + \sqrt{3})) + 13 (4 + \sqrt{3})$

Step 1.4.1.2.1.9

Expand $(4 + \sqrt{3}) (4 + \sqrt{3})$ using the FOIL Method.

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Step 1.4.1.2.1.9.1

Apply the distributive property.

$\frac{100}{3} + 17 \sqrt{3} - 4 (4 (4 + \sqrt{3}) + \sqrt{3} (4 + \sqrt{3})) + 13 (4 + \sqrt{3})$

Step 1.4.1.2.1.9.2

Apply the distributive property.

$\frac{100}{3} + 17 \sqrt{3} - 4 (4 \cdot 4 + 4 \sqrt{3} + \sqrt{3} (4 + \sqrt{3})) + 13 (4 + \sqrt{3})$

Step 1.4.1.2.1.9.3

Apply the distributive property.

$\frac{100}{3} + 17 \sqrt{3} - 4 (4 \cdot 4 + 4 \sqrt{3} + \sqrt{3} \cdot 4 + \sqrt{3} \sqrt{3}) + 13 (4 + \sqrt{3})$

Step 1.4.1.2.1.10

Simplify and combine like terms.

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Step 1.4.1.2.1.10.1

Simplify each term.

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Step 1.4.1.2.1.10.1.1

Multiply $4$ by $4$ .

$\frac{100}{3} + 17 \sqrt{3} - 4 (16 + 4 \sqrt{3} + \sqrt{3} \cdot 4 + \sqrt{3} \sqrt{3}) + 13 (4 + \sqrt{3})$

Step 1.4.1.2.1.10.1.2

Move $4$ to the left of $\sqrt{3}$ .

$\frac{100}{3} + 17 \sqrt{3} - 4 (16 + 4 \sqrt{3} + 4 \cdot \sqrt{3} + \sqrt{3} \sqrt{3}) + 13 (4 + \sqrt{3})$

Step 1.4.1.2.1.10.1.3

Combine using the product rule for radicals.

$\frac{100}{3} + 17 \sqrt{3} - 4 (16 + 4 \sqrt{3} + 4 \sqrt{3} + \sqrt{3 \cdot 3}) + 13 (4 + \sqrt{3})$

Step 1.4.1.2.1.10.1.4

Multiply $3$ by $3$ .

$\frac{100}{3} + 17 \sqrt{3} - 4 (16 + 4 \sqrt{3} + 4 \sqrt{3} + \sqrt{9}) + 13 (4 + \sqrt{3})$

Step 1.4.1.2.1.10.1.5

Rewrite $9$ as $3^{2}$ .

$\frac{100}{3} + 17 \sqrt{3} - 4 (16 + 4 \sqrt{3} + 4 \sqrt{3} + \sqrt{3^{2}}) + 13 (4 + \sqrt{3})$

Step 1.4.1.2.1.10.1.6

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

$\frac{100}{3} + 17 \sqrt{3} - 4 (16 + 4 \sqrt{3} + 4 \sqrt{3} + 3) + 13 (4 + \sqrt{3})$

Step 1.4.1.2.1.10.2

Add $16$ and $3$ .

$\frac{100}{3} + 17 \sqrt{3} - 4 (19 + 4 \sqrt{3} + 4 \sqrt{3}) + 13 (4 + \sqrt{3})$

Step 1.4.1.2.1.10.3

Add $4 \sqrt{3}$ and $4 \sqrt{3}$ .

$\frac{100}{3} + 17 \sqrt{3} - 4 (19 + 8 \sqrt{3}) + 13 (4 + \sqrt{3})$

Step 1.4.1.2.1.11

Apply the distributive property.

$\frac{100}{3} + 17 \sqrt{3} - 4 \cdot 19 - 4 (8 \sqrt{3}) + 13 (4 + \sqrt{3})$

Step 1.4.1.2.1.12

Multiply $- 4$ by $19$ .

$\frac{100}{3} + 17 \sqrt{3} - 76 - 4 (8 \sqrt{3}) + 13 (4 + \sqrt{3})$

Step 1.4.1.2.1.13

Multiply $8$ by $- 4$ .

$\frac{100}{3} + 17 \sqrt{3} - 76 - 32 \sqrt{3} + 13 (4 + \sqrt{3})$

Step 1.4.1.2.1.14

Apply the distributive property.

$\frac{100}{3} + 17 \sqrt{3} - 76 - 32 \sqrt{3} + 13 \cdot 4 + 13 \sqrt{3}$

Step 1.4.1.2.1.15

Multiply $13$ by $4$ .

$\frac{100}{3} + 17 \sqrt{3} - 76 - 32 \sqrt{3} + 52 + 13 \sqrt{3}$

Step 1.4.1.2.2

To write $- 76$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{100}{3} - 76 \cdot \frac{3}{3} + 17 \sqrt{3} - 32 \sqrt{3} + 52 + 13 \sqrt{3}$

Step 1.4.1.2.3

Combine $- 76$ and $\frac{3}{3}$ .

$\frac{100}{3} + \frac{- 76 \cdot 3}{3} + 17 \sqrt{3} - 32 \sqrt{3} + 52 + 13 \sqrt{3}$

Step 1.4.1.2.4

Combine the numerators over the common denominator.

$\frac{100 - 76 \cdot 3}{3} + 17 \sqrt{3} - 32 \sqrt{3} + 52 + 13 \sqrt{3}$

Step 1.4.1.2.5

Simplify the numerator.

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Step 1.4.1.2.5.1

Multiply $- 76$ by $3$ .

$\frac{100 - 228}{3} + 17 \sqrt{3} - 32 \sqrt{3} + 52 + 13 \sqrt{3}$

Step 1.4.1.2.5.2

Subtract $228$ from $100$ .

$\frac{- 128}{3} + 17 \sqrt{3} - 32 \sqrt{3} + 52 + 13 \sqrt{3}$

Step 1.4.1.2.6

Move the negative in front of the fraction.

$- \frac{128}{3} + 17 \sqrt{3} - 32 \sqrt{3} + 52 + 13 \sqrt{3}$

Step 1.4.1.2.7

To write $52$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$- \frac{128}{3} + 52 \cdot \frac{3}{3} + 17 \sqrt{3} - 32 \sqrt{3} + 13 \sqrt{3}$

Step 1.4.1.2.8

Combine $52$ and $\frac{3}{3}$ .

$- \frac{128}{3} + \frac{52 \cdot 3}{3} + 17 \sqrt{3} - 32 \sqrt{3} + 13 \sqrt{3}$

Step 1.4.1.2.9

Combine the numerators over the common denominator.

$\frac{- 128 + 52 \cdot 3}{3} + 17 \sqrt{3} - 32 \sqrt{3} + 13 \sqrt{3}$

Step 1.4.1.2.10

Simplify the numerator.

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Step 1.4.1.2.10.1

Multiply $52$ by $3$ .

$\frac{- 128 + 156}{3} + 17 \sqrt{3} - 32 \sqrt{3} + 13 \sqrt{3}$

Step 1.4.1.2.10.2

Add $- 128$ and $156$ .

$\frac{28}{3} + 17 \sqrt{3} - 32 \sqrt{3} + 13 \sqrt{3}$

Step 1.4.1.2.11

Subtract $32 \sqrt{3}$ from $17 \sqrt{3}$ .

$\frac{28}{3} - 15 \sqrt{3} + 13 \sqrt{3}$

Step 1.4.1.2.12

Add $- 15 \sqrt{3}$ and $13 \sqrt{3}$ .

$\frac{28}{3} - 2 \sqrt{3}$

Step 1.4.2

Evaluate at $x = 4 - \sqrt{3}$ .

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Step 1.4.2.1

Substitute $4 - \sqrt{3}$ for $x$ .

$\frac{1}{3} \cdot {(4 - \sqrt{3})}^{3} - 4 {(4 - \sqrt{3})}^{2} + 13 (4 - \sqrt{3})$

Step 1.4.2.2

Simplify.

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Step 1.4.2.2.1

Simplify each term.

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Step 1.4.2.2.1.1

Use the Binomial Theorem.

$\frac{1}{3} \cdot (4^{3} + 3 \cdot 4^{2} (- \sqrt{3}) + 3 \cdot 4 {(- \sqrt{3})}^{2} + {(- \sqrt{3})}^{3}) - 4 {(4 - \sqrt{3})}^{2} + 13 (4 - \sqrt{3})$

Step 1.4.2.2.1.2

Simplify each term.

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Step 1.4.2.2.1.2.1

Raise $4$ to the power of $3$ .

$\frac{1}{3} \cdot (64 + 3 \cdot 4^{2} (- \sqrt{3}) + 3 \cdot 4 {(- \sqrt{3})}^{2} + {(- \sqrt{3})}^{3}) - 4 {(4 - \sqrt{3})}^{2} + 13 (4 - \sqrt{3})$

Step 1.4.2.2.1.2.2

Raise $4$ to the power of $2$ .

$\frac{1}{3} \cdot (64 + 3 \cdot 16 (- \sqrt{3}) + 3 \cdot 4 {(- \sqrt{3})}^{2} + {(- \sqrt{3})}^{3}) - 4 {(4 - \sqrt{3})}^{2} + 13 (4 - \sqrt{3})$

Step 1.4.2.2.1.2.3

Multiply $3$ by $16$ .

$\frac{1}{3} \cdot (64 + 48 (- \sqrt{3}) + 3 \cdot 4 {(- \sqrt{3})}^{2} + {(- \sqrt{3})}^{3}) - 4 {(4 - \sqrt{3})}^{2} + 13 (4 - \sqrt{3})$

Step 1.4.2.2.1.2.4

Multiply $- 1$ by $48$ .

$\frac{1}{3} \cdot (64 - 48 \sqrt{3} + 3 \cdot 4 {(- \sqrt{3})}^{2} + {(- \sqrt{3})}^{3}) - 4 {(4 - \sqrt{3})}^{2} + 13 (4 - \sqrt{3})$

Step 1.4.2.2.1.2.5

Multiply $3$ by $4$ .

$\frac{1}{3} \cdot (64 - 48 \sqrt{3} + 12 {(- \sqrt{3})}^{2} + {(- \sqrt{3})}^{3}) - 4 {(4 - \sqrt{3})}^{2} + 13 (4 - \sqrt{3})$

Step 1.4.2.2.1.2.6

Apply the product rule to $- \sqrt{3}$ .

$\frac{1}{3} \cdot (64 - 48 \sqrt{3} + 12 ({(- 1)}^{2} {\sqrt{3}}^{2}) + {(- \sqrt{3})}^{3}) - 4 {(4 - \sqrt{3})}^{2} + 13 (4 - \sqrt{3})$

Step 1.4.2.2.1.2.7

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

$\frac{1}{3} \cdot (64 - 48 \sqrt{3} + 12 (1 {\sqrt{3}}^{2}) + {(- \sqrt{3})}^{3}) - 4 {(4 - \sqrt{3})}^{2} + 13 (4 - \sqrt{3})$

Step 1.4.2.2.1.2.8

Multiply ${\sqrt{3}}^{2}$ by $1$ .

$\frac{1}{3} \cdot (64 - 48 \sqrt{3} + 12 {\sqrt{3}}^{2} + {(- \sqrt{3})}^{3}) - 4 {(4 - \sqrt{3})}^{2} + 13 (4 - \sqrt{3})$

Step 1.4.2.2.1.2.9

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 1.4.2.2.1.2.9.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$\frac{1}{3} \cdot (64 - 48 \sqrt{3} + 12 {(3^{\frac{1}{2}})}^{2} + {(- \sqrt{3})}^{3}) - 4 {(4 - \sqrt{3})}^{2} + 13 (4 - \sqrt{3})$

Step 1.4.2.2.1.2.9.2

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

$\frac{1}{3} \cdot (64 - 48 \sqrt{3} + 12 \cdot 3^{\frac{1}{2} \cdot 2} + {(- \sqrt{3})}^{3}) - 4 {(4 - \sqrt{3})}^{2} + 13 (4 - \sqrt{3})$

Step 1.4.2.2.1.2.9.3

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

$\frac{1}{3} \cdot (64 - 48 \sqrt{3} + 12 \cdot 3^{\frac{2}{2}} + {(- \sqrt{3})}^{3}) - 4 {(4 - \sqrt{3})}^{2} + 13 (4 - \sqrt{3})$

Step 1.4.2.2.1.2.9.4

Cancel the common factor of $2$ .

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Step 1.4.2.2.1.2.9.4.1

Cancel the common factor.

$\frac{1}{3} \cdot (64 - 48 \sqrt{3} + 12 \cdot 3^{\frac{2}{2}} + {(- \sqrt{3})}^{3}) - 4 {(4 - \sqrt{3})}^{2} + 13 (4 - \sqrt{3})$

Step 1.4.2.2.1.2.9.4.2

Rewrite the expression.

$\frac{1}{3} \cdot (64 - 48 \sqrt{3} + 12 \cdot 3^{1} + {(- \sqrt{3})}^{3}) - 4 {(4 - \sqrt{3})}^{2} + 13 (4 - \sqrt{3})$

Step 1.4.2.2.1.2.9.5

Evaluate the exponent.

$\frac{1}{3} \cdot (64 - 48 \sqrt{3} + 12 \cdot 3 + {(- \sqrt{3})}^{3}) - 4 {(4 - \sqrt{3})}^{2} + 13 (4 - \sqrt{3})$

Step 1.4.2.2.1.2.10

Multiply $12$ by $3$ .

$\frac{1}{3} \cdot (64 - 48 \sqrt{3} + 36 + {(- \sqrt{3})}^{3}) - 4 {(4 - \sqrt{3})}^{2} + 13 (4 - \sqrt{3})$

Step 1.4.2.2.1.2.11

Apply the product rule to $- \sqrt{3}$ .

$\frac{1}{3} \cdot (64 - 48 \sqrt{3} + 36 + {(- 1)}^{3} {\sqrt{3}}^{3}) - 4 {(4 - \sqrt{3})}^{2} + 13 (4 - \sqrt{3})$

Step 1.4.2.2.1.2.12

Raise $- 1$ to the power of $3$ .

$\frac{1}{3} \cdot (64 - 48 \sqrt{3} + 36 - {\sqrt{3}}^{3}) - 4 {(4 - \sqrt{3})}^{2} + 13 (4 - \sqrt{3})$

Step 1.4.2.2.1.2.13

Rewrite ${\sqrt{3}}^{3}$ as $\sqrt{3^{3}}$ .

$\frac{1}{3} \cdot (64 - 48 \sqrt{3} + 36 - \sqrt{3^{3}}) - 4 {(4 - \sqrt{3})}^{2} + 13 (4 - \sqrt{3})$

Step 1.4.2.2.1.2.14

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

$\frac{1}{3} \cdot (64 - 48 \sqrt{3} + 36 - \sqrt{27}) - 4 {(4 - \sqrt{3})}^{2} + 13 (4 - \sqrt{3})$

Step 1.4.2.2.1.2.15

Rewrite $27$ as $3^{2} \cdot 3$ .

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Step 1.4.2.2.1.2.15.1

Factor $9$ out of $27$ .

$\frac{1}{3} \cdot (64 - 48 \sqrt{3} + 36 - \sqrt{9 (3)}) - 4 {(4 - \sqrt{3})}^{2} + 13 (4 - \sqrt{3})$

Step 1.4.2.2.1.2.15.2

Rewrite $9$ as $3^{2}$ .

$\frac{1}{3} \cdot (64 - 48 \sqrt{3} + 36 - \sqrt{3^{2} \cdot 3}) - 4 {(4 - \sqrt{3})}^{2} + 13 (4 - \sqrt{3})$

Step 1.4.2.2.1.2.16

Pull terms out from under the radical.

$\frac{1}{3} \cdot (64 - 48 \sqrt{3} + 36 - (3 \sqrt{3})) - 4 {(4 - \sqrt{3})}^{2} + 13 (4 - \sqrt{3})$

Step 1.4.2.2.1.2.17

Multiply $3$ by $- 1$ .

$\frac{1}{3} \cdot (64 - 48 \sqrt{3} + 36 - 3 \sqrt{3}) - 4 {(4 - \sqrt{3})}^{2} + 13 (4 - \sqrt{3})$

Step 1.4.2.2.1.3

Add $64$ and $36$ .

$\frac{1}{3} \cdot (100 - 48 \sqrt{3} - 3 \sqrt{3}) - 4 {(4 - \sqrt{3})}^{2} + 13 (4 - \sqrt{3})$

Step 1.4.2.2.1.4

Subtract $3 \sqrt{3}$ from $- 48 \sqrt{3}$ .

$\frac{1}{3} \cdot (100 - 51 \sqrt{3}) - 4 {(4 - \sqrt{3})}^{2} + 13 (4 - \sqrt{3})$

Step 1.4.2.2.1.5

Apply the distributive property.

$\frac{1}{3} \cdot 100 + \frac{1}{3} (- 51 \sqrt{3}) - 4 {(4 - \sqrt{3})}^{2} + 13 (4 - \sqrt{3})$

Step 1.4.2.2.1.6

Combine $\frac{1}{3}$ and $100$ .

$\frac{100}{3} + \frac{1}{3} (- 51 \sqrt{3}) - 4 {(4 - \sqrt{3})}^{2} + 13 (4 - \sqrt{3})$

Step 1.4.2.2.1.7

Cancel the common factor of $3$ .

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Step 1.4.2.2.1.7.1

Factor $3$ out of $- 51 \sqrt{3}$ .

$\frac{100}{3} + \frac{1}{3} (3 (- 17 \sqrt{3})) - 4 {(4 - \sqrt{3})}^{2} + 13 (4 - \sqrt{3})$

Step 1.4.2.2.1.7.2

Cancel the common factor.

$\frac{100}{3} + \frac{1}{3} (3 (- 17 \sqrt{3})) - 4 {(4 - \sqrt{3})}^{2} + 13 (4 - \sqrt{3})$

Step 1.4.2.2.1.7.3

Rewrite the expression.

$\frac{100}{3} - 17 \sqrt{3} - 4 {(4 - \sqrt{3})}^{2} + 13 (4 - \sqrt{3})$

Step 1.4.2.2.1.8

Rewrite ${(4 - \sqrt{3})}^{2}$ as $(4 - \sqrt{3}) (4 - \sqrt{3})$ .

$\frac{100}{3} - 17 \sqrt{3} - 4 ((4 - \sqrt{3}) (4 - \sqrt{3})) + 13 (4 - \sqrt{3})$

Step 1.4.2.2.1.9

Expand $(4 - \sqrt{3}) (4 - \sqrt{3})$ using the FOIL Method.

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Step 1.4.2.2.1.9.1

Apply the distributive property.

$\frac{100}{3} - 17 \sqrt{3} - 4 (4 (4 - \sqrt{3}) - \sqrt{3} (4 - \sqrt{3})) + 13 (4 - \sqrt{3})$

Step 1.4.2.2.1.9.2

Apply the distributive property.

$\frac{100}{3} - 17 \sqrt{3} - 4 (4 \cdot 4 + 4 (- \sqrt{3}) - \sqrt{3} (4 - \sqrt{3})) + 13 (4 - \sqrt{3})$

Step 1.4.2.2.1.9.3

Apply the distributive property.

$\frac{100}{3} - 17 \sqrt{3} - 4 (4 \cdot 4 + 4 (- \sqrt{3}) - \sqrt{3} \cdot 4 - \sqrt{3} (- \sqrt{3})) + 13 (4 - \sqrt{3})$

Step 1.4.2.2.1.10

Simplify and combine like terms.

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Step 1.4.2.2.1.10.1

Simplify each term.

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Step 1.4.2.2.1.10.1.1

Multiply $4$ by $4$ .

$\frac{100}{3} - 17 \sqrt{3} - 4 (16 + 4 (- \sqrt{3}) - \sqrt{3} \cdot 4 - \sqrt{3} (- \sqrt{3})) + 13 (4 - \sqrt{3})$

Step 1.4.2.2.1.10.1.2

Multiply $- 1$ by $4$ .

$\frac{100}{3} - 17 \sqrt{3} - 4 (16 - 4 \sqrt{3} - \sqrt{3} \cdot 4 - \sqrt{3} (- \sqrt{3})) + 13 (4 - \sqrt{3})$

Step 1.4.2.2.1.10.1.3

Multiply $4$ by $- 1$ .

$\frac{100}{3} - 17 \sqrt{3} - 4 (16 - 4 \sqrt{3} - 4 \sqrt{3} - \sqrt{3} (- \sqrt{3})) + 13 (4 - \sqrt{3})$

Step 1.4.2.2.1.10.1.4

Multiply $- \sqrt{3} (- \sqrt{3})$ .

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Step 1.4.2.2.1.10.1.4.1

Multiply $- 1$ by $- 1$ .

$\frac{100}{3} - 17 \sqrt{3} - 4 (16 - 4 \sqrt{3} - 4 \sqrt{3} + 1 \sqrt{3} \sqrt{3}) + 13 (4 - \sqrt{3})$

Step 1.4.2.2.1.10.1.4.2

Multiply $\sqrt{3}$ by $1$ .

$\frac{100}{3} - 17 \sqrt{3} - 4 (16 - 4 \sqrt{3} - 4 \sqrt{3} + \sqrt{3} \sqrt{3}) + 13 (4 - \sqrt{3})$

Step 1.4.2.2.1.10.1.4.3

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

$\frac{100}{3} - 17 \sqrt{3} - 4 (16 - 4 \sqrt{3} - 4 \sqrt{3} + {\sqrt{3}}^{1} \sqrt{3}) + 13 (4 - \sqrt{3})$

Step 1.4.2.2.1.10.1.4.4

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

$\frac{100}{3} - 17 \sqrt{3} - 4 (16 - 4 \sqrt{3} - 4 \sqrt{3} + {\sqrt{3}}^{1} {\sqrt{3}}^{1}) + 13 (4 - \sqrt{3})$

Step 1.4.2.2.1.10.1.4.5

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

$\frac{100}{3} - 17 \sqrt{3} - 4 (16 - 4 \sqrt{3} - 4 \sqrt{3} + {\sqrt{3}}^{1 + 1}) + 13 (4 - \sqrt{3})$

Step 1.4.2.2.1.10.1.4.6

Add $1$ and $1$ .

$\frac{100}{3} - 17 \sqrt{3} - 4 (16 - 4 \sqrt{3} - 4 \sqrt{3} + {\sqrt{3}}^{2}) + 13 (4 - \sqrt{3})$

Step 1.4.2.2.1.10.1.5

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 1.4.2.2.1.10.1.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$\frac{100}{3} - 17 \sqrt{3} - 4 (16 - 4 \sqrt{3} - 4 \sqrt{3} + {(3^{\frac{1}{2}})}^{2}) + 13 (4 - \sqrt{3})$

Step 1.4.2.2.1.10.1.5.2

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

$\frac{100}{3} - 17 \sqrt{3} - 4 (16 - 4 \sqrt{3} - 4 \sqrt{3} + 3^{\frac{1}{2} \cdot 2}) + 13 (4 - \sqrt{3})$

Step 1.4.2.2.1.10.1.5.3

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

$\frac{100}{3} - 17 \sqrt{3} - 4 (16 - 4 \sqrt{3} - 4 \sqrt{3} + 3^{\frac{2}{2}}) + 13 (4 - \sqrt{3})$

Step 1.4.2.2.1.10.1.5.4

Cancel the common factor of $2$ .

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Step 1.4.2.2.1.10.1.5.4.1

Cancel the common factor.

$\frac{100}{3} - 17 \sqrt{3} - 4 (16 - 4 \sqrt{3} - 4 \sqrt{3} + 3^{\frac{2}{2}}) + 13 (4 - \sqrt{3})$

Step 1.4.2.2.1.10.1.5.4.2

Rewrite the expression.

$\frac{100}{3} - 17 \sqrt{3} - 4 (16 - 4 \sqrt{3} - 4 \sqrt{3} + 3^{1}) + 13 (4 - \sqrt{3})$

Step 1.4.2.2.1.10.1.5.5

Evaluate the exponent.

$\frac{100}{3} - 17 \sqrt{3} - 4 (16 - 4 \sqrt{3} - 4 \sqrt{3} + 3) + 13 (4 - \sqrt{3})$

Step 1.4.2.2.1.10.2

Add $16$ and $3$ .

$\frac{100}{3} - 17 \sqrt{3} - 4 (19 - 4 \sqrt{3} - 4 \sqrt{3}) + 13 (4 - \sqrt{3})$

Step 1.4.2.2.1.10.3

Subtract $4 \sqrt{3}$ from $- 4 \sqrt{3}$ .

$\frac{100}{3} - 17 \sqrt{3} - 4 (19 - 8 \sqrt{3}) + 13 (4 - \sqrt{3})$

Step 1.4.2.2.1.11

Apply the distributive property.

$\frac{100}{3} - 17 \sqrt{3} - 4 \cdot 19 - 4 (- 8 \sqrt{3}) + 13 (4 - \sqrt{3})$

Step 1.4.2.2.1.12

Multiply $- 4$ by $19$ .

$\frac{100}{3} - 17 \sqrt{3} - 76 - 4 (- 8 \sqrt{3}) + 13 (4 - \sqrt{3})$

Step 1.4.2.2.1.13

Multiply $- 8$ by $- 4$ .

$\frac{100}{3} - 17 \sqrt{3} - 76 + 32 \sqrt{3} + 13 (4 - \sqrt{3})$

Step 1.4.2.2.1.14

Apply the distributive property.

$\frac{100}{3} - 17 \sqrt{3} - 76 + 32 \sqrt{3} + 13 \cdot 4 + 13 (- \sqrt{3})$

Step 1.4.2.2.1.15

Multiply $13$ by $4$ .

$\frac{100}{3} - 17 \sqrt{3} - 76 + 32 \sqrt{3} + 52 + 13 (- \sqrt{3})$

Step 1.4.2.2.1.16

Multiply $- 1$ by $13$ .

$\frac{100}{3} - 17 \sqrt{3} - 76 + 32 \sqrt{3} + 52 - 13 \sqrt{3}$

Step 1.4.2.2.2

To write $- 76$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{100}{3} - 76 \cdot \frac{3}{3} - 17 \sqrt{3} + 32 \sqrt{3} + 52 - 13 \sqrt{3}$

Step 1.4.2.2.3

Combine $- 76$ and $\frac{3}{3}$ .

$\frac{100}{3} + \frac{- 76 \cdot 3}{3} - 17 \sqrt{3} + 32 \sqrt{3} + 52 - 13 \sqrt{3}$

Step 1.4.2.2.4

Combine the numerators over the common denominator.

$\frac{100 - 76 \cdot 3}{3} - 17 \sqrt{3} + 32 \sqrt{3} + 52 - 13 \sqrt{3}$

Step 1.4.2.2.5

Simplify the numerator.

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Step 1.4.2.2.5.1

Multiply $- 76$ by $3$ .

$\frac{100 - 228}{3} - 17 \sqrt{3} + 32 \sqrt{3} + 52 - 13 \sqrt{3}$

Step 1.4.2.2.5.2

Subtract $228$ from $100$ .

$\frac{- 128}{3} - 17 \sqrt{3} + 32 \sqrt{3} + 52 - 13 \sqrt{3}$

Step 1.4.2.2.6

Move the negative in front of the fraction.

$- \frac{128}{3} - 17 \sqrt{3} + 32 \sqrt{3} + 52 - 13 \sqrt{3}$

Step 1.4.2.2.7

To write $52$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$- \frac{128}{3} + 52 \cdot \frac{3}{3} - 17 \sqrt{3} + 32 \sqrt{3} - 13 \sqrt{3}$

Step 1.4.2.2.8

Combine $52$ and $\frac{3}{3}$ .

$- \frac{128}{3} + \frac{52 \cdot 3}{3} - 17 \sqrt{3} + 32 \sqrt{3} - 13 \sqrt{3}$

Step 1.4.2.2.9

Combine the numerators over the common denominator.

$\frac{- 128 + 52 \cdot 3}{3} - 17 \sqrt{3} + 32 \sqrt{3} - 13 \sqrt{3}$

Step 1.4.2.2.10

Simplify the numerator.

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Step 1.4.2.2.10.1

Multiply $52$ by $3$ .

$\frac{- 128 + 156}{3} - 17 \sqrt{3} + 32 \sqrt{3} - 13 \sqrt{3}$

Step 1.4.2.2.10.2

Add $- 128$ and $156$ .

$\frac{28}{3} - 17 \sqrt{3} + 32 \sqrt{3} - 13 \sqrt{3}$

Step 1.4.2.2.11

Add $- 17 \sqrt{3}$ and $32 \sqrt{3}$ .

$\frac{28}{3} + 15 \sqrt{3} - 13 \sqrt{3}$

Step 1.4.2.2.12

Subtract $13 \sqrt{3}$ from $15 \sqrt{3}$ .

$\frac{28}{3} + 2 \sqrt{3}$

Step 1.4.3

List all of the points.

$(4 + \sqrt{3}, \frac{28}{3} - 2 \sqrt{3}), (4 - \sqrt{3}, \frac{28}{3} + 2 \sqrt{3})$

Step 2

Evaluate at the included endpoints.

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Step 2.1

Evaluate at $x = 1$ .

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Step 2.1.1

Substitute $1$ for $x$ .

$\frac{1}{3} \cdot {(1)}^{3} - 4 {(1)}^{2} + 13 (1)$

Step 2.1.2

Simplify.

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Step 2.1.2.1

Simplify each term.

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Step 2.1.2.1.1

One to any power is one.

$\frac{1}{3} \cdot 1 - 4 {(1)}^{2} + 13 (1)$

Step 2.1.2.1.2

Multiply $\frac{1}{3}$ by $1$ .

$\frac{1}{3} - 4 {(1)}^{2} + 13 (1)$

Step 2.1.2.1.3

One to any power is one.

$\frac{1}{3} - 4 \cdot 1 + 13 (1)$

Step 2.1.2.1.4

Multiply $- 4$ by $1$ .

$\frac{1}{3} - 4 + 13 (1)$

Step 2.1.2.1.5

Multiply $13$ by $1$ .

$\frac{1}{3} - 4 + 13$

Step 2.1.2.2

Find the common denominator.

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Step 2.1.2.2.1

Write $- 4$ as a fraction with denominator $1$ .

$\frac{1}{3} + \frac{- 4}{1} + 13$

Step 2.1.2.2.2

Multiply $\frac{- 4}{1}$ by $\frac{3}{3}$ .

$\frac{1}{3} + \frac{- 4}{1} \cdot \frac{3}{3} + 13$

Step 2.1.2.2.3

Multiply $\frac{- 4}{1}$ by $\frac{3}{3}$ .

$\frac{1}{3} + \frac{- 4 \cdot 3}{3} + 13$

Step 2.1.2.2.4

Write $13$ as a fraction with denominator $1$ .

$\frac{1}{3} + \frac{- 4 \cdot 3}{3} + \frac{13}{1}$

Step 2.1.2.2.5

Multiply $\frac{13}{1}$ by $\frac{3}{3}$ .

$\frac{1}{3} + \frac{- 4 \cdot 3}{3} + \frac{13}{1} \cdot \frac{3}{3}$

Step 2.1.2.2.6

Multiply $\frac{13}{1}$ by $\frac{3}{3}$ .

$\frac{1}{3} + \frac{- 4 \cdot 3}{3} + \frac{13 \cdot 3}{3}$

Step 2.1.2.3

Combine the numerators over the common denominator.

$\frac{1 - 4 \cdot 3 + 13 \cdot 3}{3}$

Step 2.1.2.4

Simplify each term.

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Step 2.1.2.4.1

Multiply $- 4$ by $3$ .

$\frac{1 - 12 + 13 \cdot 3}{3}$

Step 2.1.2.4.2

Multiply $13$ by $3$ .

$\frac{1 - 12 + 39}{3}$

Step 2.1.2.5

Simplify by adding and subtracting.

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Step 2.1.2.5.1

Subtract $12$ from $1$ .

$\frac{- 11 + 39}{3}$

Step 2.1.2.5.2

Add $- 11$ and $39$ .

$\frac{28}{3}$

Step 2.2

Evaluate at $x = 6$ .

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Step 2.2.1

Substitute $6$ for $x$ .

$\frac{1}{3} \cdot {(6)}^{3} - 4 {(6)}^{2} + 13 (6)$

Step 2.2.2

Simplify.

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Step 2.2.2.1

Simplify each term.

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Step 2.2.2.1.1

Raise $6$ to the power of $3$ .

$\frac{1}{3} \cdot 216 - 4 {(6)}^{2} + 13 (6)$

Step 2.2.2.1.2

Cancel the common factor of $3$ .

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Step 2.2.2.1.2.1

Factor $3$ out of $216$ .

$\frac{1}{3} \cdot (3 (72)) - 4 {(6)}^{2} + 13 (6)$

Step 2.2.2.1.2.2

Cancel the common factor.

$\frac{1}{3} \cdot (3 \cdot 72) - 4 {(6)}^{2} + 13 (6)$

Step 2.2.2.1.2.3

Rewrite the expression.

$72 - 4 {(6)}^{2} + 13 (6)$

Step 2.2.2.1.3

Raise $6$ to the power of $2$ .

$72 - 4 \cdot 36 + 13 (6)$

Step 2.2.2.1.4

Multiply $- 4$ by $36$ .

$72 - 144 + 13 (6)$

Step 2.2.2.1.5

Multiply $13$ by $6$ .

$72 - 144 + 78$

Step 2.2.2.2

Simplify by adding and subtracting.

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Step 2.2.2.2.1

Subtract $144$ from $72$ .

$- 72 + 78$

Step 2.2.2.2.2

Add $- 72$ and $78$ .

$6$

Step 2.3

List all of the points.

$(1, \frac{28}{3}), (6, 6)$

Step 3

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: $(4 - \sqrt{3}, \frac{28}{3} + 2 \sqrt{3})$

Absolute Minimum: $(4 + \sqrt{3}, \frac{28}{3} - 2 \sqrt{3})$

Step 4