Algebra Examples

$x^{3} - 7 x^{2} + 10 x$

Step 1

Write $x^{3} - 7 x^{2} + 10 x$ as a function.

$f (x) = x^{3} - 7 x^{2} + 10 x$

Step 2

Find the first derivative of the function.

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Differentiate.

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By the Sum Rule, the derivative of $x^{3} - 7 x^{2} + 10 x$ with respect to $x$ is $\frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 7 x^{2}] + \frac{d}{d x} [10 x]$ .

$\frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 7 x^{2}] + \frac{d}{d x} [10 x]$

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

$3 x^{2} + \frac{d}{d x} [- 7 x^{2}] + \frac{d}{d x} [10 x]$

Evaluate $\frac{d}{d x} [- 7 x^{2}]$ .

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Since $- 7$ is constant with respect to $x$ , the derivative of $- 7 x^{2}$ with respect to $x$ is $- 7 \frac{d}{d x} [x^{2}]$ .

$3 x^{2} - 7 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [10 x]$

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

$3 x^{2} - 7 (2 x) + \frac{d}{d x} [10 x]$

Multiply $2$ by $- 7$ .

$3 x^{2} - 14 x + \frac{d}{d x} [10 x]$

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

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

$3 x^{2} - 14 x + 10 \frac{d}{d x} [x]$

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

$3 x^{2} - 14 x + 10 \cdot 1$

Multiply $10$ by $1$ .

$3 x^{2} - 14 x + 10$

Step 3

Find the second derivative of the function.

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By the Sum Rule, the derivative of $3 x^{2} - 14 x + 10$ with respect to $x$ is $\frac{d}{d x} [3 x^{2}] + \frac{d}{d x} [- 14 x] + \frac{d}{d x} [10]$ .

$f'' (x) = \frac{d}{d x} (3 x^{2}) + \frac{d}{d x} (- 14 x) + \frac{d}{d x} (10)$

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

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

$f'' (x) = 3 \frac{d}{d x} (x^{2}) + \frac{d}{d x} (- 14 x) + \frac{d}{d x} (10)$

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

$f'' (x) = 3 (2 x) + \frac{d}{d x} (- 14 x) + \frac{d}{d x} (10)$

Multiply $2$ by $3$ .

$f'' (x) = 6 x + \frac{d}{d x} (- 14 x) + \frac{d}{d x} (10)$

Evaluate $\frac{d}{d x} [- 14 x]$ .

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Since $- 14$ is constant with respect to $x$ , the derivative of $- 14 x$ with respect to $x$ is $- 14 \frac{d}{d x} [x]$ .

$f'' (x) = 6 x - 14 \frac{d}{d x} x + \frac{d}{d x} (10)$

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

$f'' (x) = 6 x - 14 \cdot 1 + \frac{d}{d x} (10)$

Multiply $- 14$ by $1$ .

$f'' (x) = 6 x - 14 + \frac{d}{d x} (10)$

Differentiate using the Constant Rule.

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Since $10$ is constant with respect to $x$ , the derivative of $10$ with respect to $x$ is $0$ .

$f'' (x) = 6 x - 14 + 0$

Add $6 x - 14$ and $0$ .

$f'' (x) = 6 x - 14$

Step 4

To find the local maximum and minimum values of the function, set the derivative equal to $0$ and solve.

$3 x^{2} - 14 x + 10 = 0$

Step 5

Find the first derivative.

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Find the first derivative.

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Differentiate.

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By the Sum Rule, the derivative of $x^{3} - 7 x^{2} + 10 x$ with respect to $x$ is $\frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 7 x^{2}] + \frac{d}{d x} [10 x]$ .

$f' (x) = \frac{d}{d x} (x^{3}) + \frac{d}{d x} (- 7 x^{2}) + \frac{d}{d x} (10 x)$

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

$f' (x) = 3 x^{2} + \frac{d}{d x} (- 7 x^{2}) + \frac{d}{d x} (10 x)$

Evaluate $\frac{d}{d x} [- 7 x^{2}]$ .

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Since $- 7$ is constant with respect to $x$ , the derivative of $- 7 x^{2}$ with respect to $x$ is $- 7 \frac{d}{d x} [x^{2}]$ .

$f' (x) = 3 x^{2} - 7 \frac{d}{d x} x^{2} + \frac{d}{d x} (10 x)$

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

$f' (x) = 3 x^{2} - 7 (2 x) + \frac{d}{d x} (10 x)$

Multiply $2$ by $- 7$ .

$f' (x) = 3 x^{2} - 14 x + \frac{d}{d x} (10 x)$

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

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

$f' (x) = 3 x^{2} - 14 x + 10 \frac{d}{d x} (x)$

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

$f' (x) = 3 x^{2} - 14 x + 10 \cdot 1$

Multiply $10$ by $1$ .

$f' (x) = 3 x^{2} - 14 x + 10$

The first derivative of $f (x)$ with respect to $x$ is $3 x^{2} - 14 x + 10$ .

$3 x^{2} - 14 x + 10$

Step 6

Set the first derivative equal to $0$ then solve the equation $3 x^{2} - 14 x + 10 = 0$ .

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Set the first derivative equal to $0$ .

$3 x^{2} - 14 x + 10 = 0$

Use the quadratic formula to find the solutions.

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

Substitute the values $a = 3$ , $b = - 14$ , and $c = 10$ into the quadratic formula and solve for $x$ .

$\frac{14 \pm \sqrt{{(- 14)}^{2} - 4 \cdot (3 \cdot 10)}}{2 \cdot 3}$

Simplify.

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Simplify the numerator.

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Raise $- 14$ to the power of $2$ .

$x = \frac{14 \pm \sqrt{196 - 4 \cdot 3 \cdot 10}}{2 \cdot 3}$

Multiply $- 4 \cdot 3 \cdot 10$ .

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Multiply $- 4$ by $3$ .

$x = \frac{14 \pm \sqrt{196 - 12 \cdot 10}}{2 \cdot 3}$

Multiply $- 12$ by $10$ .

$x = \frac{14 \pm \sqrt{196 - 120}}{2 \cdot 3}$

Subtract $120$ from $196$ .

$x = \frac{14 \pm \sqrt{76}}{2 \cdot 3}$

Rewrite $76$ as $2^{2} \cdot 19$ .

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Factor $4$ out of $76$ .

$x = \frac{14 \pm \sqrt{4 (19)}}{2 \cdot 3}$

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

$x = \frac{14 \pm \sqrt{2^{2} \cdot 19}}{2 \cdot 3}$

Pull terms out from under the radical.

$x = \frac{14 \pm 2 \sqrt{19}}{2 \cdot 3}$

Multiply $2$ by $3$ .

$x = \frac{14 \pm 2 \sqrt{19}}{6}$

Simplify $\frac{14 \pm 2 \sqrt{19}}{6}$ .

$x = \frac{7 \pm \sqrt{19}}{3}$

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

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Simplify the numerator.

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Raise $- 14$ to the power of $2$ .

$x = \frac{14 \pm \sqrt{196 - 4 \cdot 3 \cdot 10}}{2 \cdot 3}$

Multiply $- 4 \cdot 3 \cdot 10$ .

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Multiply $- 4$ by $3$ .

$x = \frac{14 \pm \sqrt{196 - 12 \cdot 10}}{2 \cdot 3}$

Multiply $- 12$ by $10$ .

$x = \frac{14 \pm \sqrt{196 - 120}}{2 \cdot 3}$

Subtract $120$ from $196$ .

$x = \frac{14 \pm \sqrt{76}}{2 \cdot 3}$

Rewrite $76$ as $2^{2} \cdot 19$ .

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Factor $4$ out of $76$ .

$x = \frac{14 \pm \sqrt{4 (19)}}{2 \cdot 3}$

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

$x = \frac{14 \pm \sqrt{2^{2} \cdot 19}}{2 \cdot 3}$

Pull terms out from under the radical.

$x = \frac{14 \pm 2 \sqrt{19}}{2 \cdot 3}$

Multiply $2$ by $3$ .

$x = \frac{14 \pm 2 \sqrt{19}}{6}$

Simplify $\frac{14 \pm 2 \sqrt{19}}{6}$ .

$x = \frac{7 \pm \sqrt{19}}{3}$

Change the $\pm$ to $+$ .

$x = \frac{7 + \sqrt{19}}{3}$

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

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Simplify the numerator.

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Raise $- 14$ to the power of $2$ .

$x = \frac{14 \pm \sqrt{196 - 4 \cdot 3 \cdot 10}}{2 \cdot 3}$

Multiply $- 4 \cdot 3 \cdot 10$ .

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Multiply $- 4$ by $3$ .

$x = \frac{14 \pm \sqrt{196 - 12 \cdot 10}}{2 \cdot 3}$

Multiply $- 12$ by $10$ .

$x = \frac{14 \pm \sqrt{196 - 120}}{2 \cdot 3}$

Subtract $120$ from $196$ .

$x = \frac{14 \pm \sqrt{76}}{2 \cdot 3}$

Rewrite $76$ as $2^{2} \cdot 19$ .

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Factor $4$ out of $76$ .

$x = \frac{14 \pm \sqrt{4 (19)}}{2 \cdot 3}$

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

$x = \frac{14 \pm \sqrt{2^{2} \cdot 19}}{2 \cdot 3}$

Pull terms out from under the radical.

$x = \frac{14 \pm 2 \sqrt{19}}{2 \cdot 3}$

Multiply $2$ by $3$ .

$x = \frac{14 \pm 2 \sqrt{19}}{6}$

Simplify $\frac{14 \pm 2 \sqrt{19}}{6}$ .

$x = \frac{7 \pm \sqrt{19}}{3}$

Change the $\pm$ to $-$ .

$x = \frac{7 - \sqrt{19}}{3}$

The final answer is the combination of both solutions.

$x = \frac{7 + \sqrt{19}}{3}, \frac{7 - \sqrt{19}}{3}$

Step 7

Find the values where the derivative is undefined.

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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 8

Critical points to evaluate.

$x = \frac{7 + \sqrt{19}}{3}, \frac{7 - \sqrt{19}}{3}$

Step 9

Evaluate the second derivative at $x = \frac{7 + \sqrt{19}}{3}$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$6 (\frac{7 + \sqrt{19}}{3}) - 14$

Step 10

Evaluate the second derivative.

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Simplify each term.

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Cancel the common factor of $3$ .

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Factor $3$ out of $6$ .

$3 (2) \frac{7 + \sqrt{19}}{3} - 14$

Cancel the common factor.

$3 \cdot 2 \frac{7 + \sqrt{19}}{3} - 14$

Rewrite the expression.

$2 (7 + \sqrt{19}) - 14$

Apply the distributive property.

$2 \cdot 7 + 2 \sqrt{19} - 14$

Multiply $2$ by $7$ .

$14 + 2 \sqrt{19} - 14$

Simplify by subtracting numbers.

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Subtract $14$ from $14$ .

$0 + 2 \sqrt{19}$

Add $0$ and $2 \sqrt{19}$ .

$2 \sqrt{19}$

Step 11

$x = \frac{7 + \sqrt{19}}{3}$ is a local minimum because the value of the second derivative is positive. This is referred to as the second derivative test.

$x = \frac{7 + \sqrt{19}}{3}$ is a local minimum

Step 12

Find the y-value when $x = \frac{7 + \sqrt{19}}{3}$ .

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Replace the variable $x$ with $\frac{7 + \sqrt{19}}{3}$ in the expression.

$f (\frac{7 + \sqrt{19}}{3}) = {(\frac{7 + \sqrt{19}}{3})}^{3} - 7 {(\frac{7 + \sqrt{19}}{3})}^{2} + 10 (\frac{7 + \sqrt{19}}{3})$

Simplify the result.

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Simplify each term.

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Apply the product rule to $\frac{7 + \sqrt{19}}{3}$ .

$f (\frac{7 + \sqrt{19}}{3}) = \frac{{(7 + \sqrt{19})}^{3}}{3^{3}} - 7 {(\frac{7 + \sqrt{19}}{3})}^{2} + 10 (\frac{7 + \sqrt{19}}{3})$

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

$f (\frac{7 + \sqrt{19}}{3}) = \frac{{(7 + \sqrt{19})}^{3}}{27} - 7 {(\frac{7 + \sqrt{19}}{3})}^{2} + 10 (\frac{7 + \sqrt{19}}{3})$

Use the Binomial Theorem.

$f (\frac{7 + \sqrt{19}}{3}) = \frac{7^{3} + 3 \cdot (7^{2} \sqrt{19}) + 3 \cdot (7 {\sqrt{19}}^{2}) + {\sqrt{19}}^{3}}{27} - 7 {(\frac{7 + \sqrt{19}}{3})}^{2} + 10 (\frac{7 + \sqrt{19}}{3})$

Simplify each term.

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Raise $7$ to the power of $3$ .

$f (\frac{7 + \sqrt{19}}{3}) = \frac{343 + 3 \cdot (7^{2} \sqrt{19}) + 3 \cdot (7 {\sqrt{19}}^{2}) + {\sqrt{19}}^{3}}{27} - 7 {(\frac{7 + \sqrt{19}}{3})}^{2} + 10 (\frac{7 + \sqrt{19}}{3})$

Raise $7$ to the power of $2$ .

$f (\frac{7 + \sqrt{19}}{3}) = \frac{343 + 3 \cdot (49 \sqrt{19}) + 3 \cdot (7 {\sqrt{19}}^{2}) + {\sqrt{19}}^{3}}{27} - 7 {(\frac{7 + \sqrt{19}}{3})}^{2} + 10 (\frac{7 + \sqrt{19}}{3})$

Multiply $3$ by $49$ .

$f (\frac{7 + \sqrt{19}}{3}) = \frac{343 + 147 \sqrt{19} + 3 \cdot (7 {\sqrt{19}}^{2}) + {\sqrt{19}}^{3}}{27} - 7 {(\frac{7 + \sqrt{19}}{3})}^{2} + 10 (\frac{7 + \sqrt{19}}{3})$

Multiply $3$ by $7$ .

$f (\frac{7 + \sqrt{19}}{3}) = \frac{343 + 147 \sqrt{19} + 21 {\sqrt{19}}^{2} + {\sqrt{19}}^{3}}{27} - 7 {(\frac{7 + \sqrt{19}}{3})}^{2} + 10 (\frac{7 + \sqrt{19}}{3})$

Rewrite ${\sqrt{19}}^{2}$ as $19$ .

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Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{19}$ as $19^{\frac{1}{2}}$ .

$f (\frac{7 + \sqrt{19}}{3}) = \frac{343 + 147 \sqrt{19} + 21 {(19^{\frac{1}{2}})}^{2} + {\sqrt{19}}^{3}}{27} - 7 {(\frac{7 + \sqrt{19}}{3})}^{2} + 10 (\frac{7 + \sqrt{19}}{3})$

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

$f (\frac{7 + \sqrt{19}}{3}) = \frac{343 + 147 \sqrt{19} + 21 \cdot 19^{\frac{1}{2} \cdot 2} + {\sqrt{19}}^{3}}{27} - 7 {(\frac{7 + \sqrt{19}}{3})}^{2} + 10 (\frac{7 + \sqrt{19}}{3})$

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

$f (\frac{7 + \sqrt{19}}{3}) = \frac{343 + 147 \sqrt{19} + 21 \cdot 19^{\frac{2}{2}} + {\sqrt{19}}^{3}}{27} - 7 {(\frac{7 + \sqrt{19}}{3})}^{2} + 10 (\frac{7 + \sqrt{19}}{3})$

Cancel the common factor of $2$ .

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Cancel the common factor.

$f (\frac{7 + \sqrt{19}}{3}) = \frac{343 + 147 \sqrt{19} + 21 \cdot 19^{\frac{2}{2}} + {\sqrt{19}}^{3}}{27} - 7 {(\frac{7 + \sqrt{19}}{3})}^{2} + 10 (\frac{7 + \sqrt{19}}{3})$

Rewrite the expression.

$f (\frac{7 + \sqrt{19}}{3}) = \frac{343 + 147 \sqrt{19} + 21 \cdot 19 + {\sqrt{19}}^{3}}{27} - 7 {(\frac{7 + \sqrt{19}}{3})}^{2} + 10 (\frac{7 + \sqrt{19}}{3})$

Evaluate the exponent.

$f (\frac{7 + \sqrt{19}}{3}) = \frac{343 + 147 \sqrt{19} + 21 \cdot 19 + {\sqrt{19}}^{3}}{27} - 7 {(\frac{7 + \sqrt{19}}{3})}^{2} + 10 (\frac{7 + \sqrt{19}}{3})$

Multiply $21$ by $19$ .

$f (\frac{7 + \sqrt{19}}{3}) = \frac{343 + 147 \sqrt{19} + 399 + {\sqrt{19}}^{3}}{27} - 7 {(\frac{7 + \sqrt{19}}{3})}^{2} + 10 (\frac{7 + \sqrt{19}}{3})$

Rewrite ${\sqrt{19}}^{3}$ as ${(19^{3})}^{\frac{1}{2}}$ .

$f (\frac{7 + \sqrt{19}}{3}) = \frac{343 + 147 \sqrt{19} + 399 + \sqrt{19^{3}}}{27} - 7 {(\frac{7 + \sqrt{19}}{3})}^{2} + 10 (\frac{7 + \sqrt{19}}{3})$

Raise $19$ to the power of $3$ .

$f (\frac{7 + \sqrt{19}}{3}) = \frac{343 + 147 \sqrt{19} + 399 + \sqrt{6859}}{27} - 7 {(\frac{7 + \sqrt{19}}{3})}^{2} + 10 (\frac{7 + \sqrt{19}}{3})$

Rewrite $6859$ as $19^{2} \cdot 19$ .

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Factor $361$ out of $6859$ .

$f (\frac{7 + \sqrt{19}}{3}) = \frac{343 + 147 \sqrt{19} + 399 + \sqrt{361 (19)}}{27} - 7 {(\frac{7 + \sqrt{19}}{3})}^{2} + 10 (\frac{7 + \sqrt{19}}{3})$

Rewrite $361$ as $19^{2}$ .

$f (\frac{7 + \sqrt{19}}{3}) = \frac{343 + 147 \sqrt{19} + 399 + \sqrt{19^{2} \cdot 19}}{27} - 7 {(\frac{7 + \sqrt{19}}{3})}^{2} + 10 (\frac{7 + \sqrt{19}}{3})$

Pull terms out from under the radical.

$f (\frac{7 + \sqrt{19}}{3}) = \frac{343 + 147 \sqrt{19} + 399 + 19 \sqrt{19}}{27} - 7 {(\frac{7 + \sqrt{19}}{3})}^{2} + 10 (\frac{7 + \sqrt{19}}{3})$

Add $343$ and $399$ .

$f (\frac{7 + \sqrt{19}}{3}) = \frac{742 + 147 \sqrt{19} + 19 \sqrt{19}}{27} - 7 {(\frac{7 + \sqrt{19}}{3})}^{2} + 10 (\frac{7 + \sqrt{19}}{3})$

Add $147 \sqrt{19}$ and $19 \sqrt{19}$ .

$f (\frac{7 + \sqrt{19}}{3}) = \frac{742 + 166 \sqrt{19}}{27} - 7 {(\frac{7 + \sqrt{19}}{3})}^{2} + 10 (\frac{7 + \sqrt{19}}{3})$

Apply the product rule to $\frac{7 + \sqrt{19}}{3}$ .

$f (\frac{7 + \sqrt{19}}{3}) = \frac{742 + 166 \sqrt{19}}{27} - 7 \frac{{(7 + \sqrt{19})}^{2}}{3^{2}} + 10 (\frac{7 + \sqrt{19}}{3})$

Raise $3$ to the power of $2$ .

$f (\frac{7 + \sqrt{19}}{3}) = \frac{742 + 166 \sqrt{19}}{27} - 7 \frac{{(7 + \sqrt{19})}^{2}}{9} + 10 (\frac{7 + \sqrt{19}}{3})$

Rewrite ${(7 + \sqrt{19})}^{2}$ as $(7 + \sqrt{19}) (7 + \sqrt{19})$ .

$f (\frac{7 + \sqrt{19}}{3}) = \frac{742 + 166 \sqrt{19}}{27} - 7 \frac{(7 + \sqrt{19}) (7 + \sqrt{19})}{9} + 10 (\frac{7 + \sqrt{19}}{3})$

Expand $(7 + \sqrt{19}) (7 + \sqrt{19})$ using the FOIL Method.

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Apply the distributive property.

$f (\frac{7 + \sqrt{19}}{3}) = \frac{742 + 166 \sqrt{19}}{27} - 7 \frac{7 (7 + \sqrt{19}) + \sqrt{19} (7 + \sqrt{19})}{9} + 10 (\frac{7 + \sqrt{19}}{3})$

Apply the distributive property.

$f (\frac{7 + \sqrt{19}}{3}) = \frac{742 + 166 \sqrt{19}}{27} - 7 \frac{7 \cdot 7 + 7 \sqrt{19} + \sqrt{19} (7 + \sqrt{19})}{9} + 10 (\frac{7 + \sqrt{19}}{3})$

Apply the distributive property.

$f (\frac{7 + \sqrt{19}}{3}) = \frac{742 + 166 \sqrt{19}}{27} - 7 \frac{7 \cdot 7 + 7 \sqrt{19} + \sqrt{19} \cdot 7 + \sqrt{19} \sqrt{19}}{9} + 10 (\frac{7 + \sqrt{19}}{3})$

Simplify and combine like terms.

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Simplify each term.

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Multiply $7$ by $7$ .

$f (\frac{7 + \sqrt{19}}{3}) = \frac{742 + 166 \sqrt{19}}{27} - 7 \frac{49 + 7 \sqrt{19} + \sqrt{19} \cdot 7 + \sqrt{19} \sqrt{19}}{9} + 10 (\frac{7 + \sqrt{19}}{3})$

Move $7$ to the left of $\sqrt{19}$ .

$f (\frac{7 + \sqrt{19}}{3}) = \frac{742 + 166 \sqrt{19}}{27} - 7 \frac{49 + 7 \sqrt{19} + 7 \cdot \sqrt{19} + \sqrt{19} \sqrt{19}}{9} + 10 (\frac{7 + \sqrt{19}}{3})$

Combine using the product rule for radicals.

$f (\frac{7 + \sqrt{19}}{3}) = \frac{742 + 166 \sqrt{19}}{27} - 7 \frac{49 + 7 \sqrt{19} + 7 \sqrt{19} + \sqrt{19 \cdot 19}}{9} + 10 (\frac{7 + \sqrt{19}}{3})$

Multiply $19$ by $19$ .

$f (\frac{7 + \sqrt{19}}{3}) = \frac{742 + 166 \sqrt{19}}{27} - 7 \frac{49 + 7 \sqrt{19} + 7 \sqrt{19} + \sqrt{361}}{9} + 10 (\frac{7 + \sqrt{19}}{3})$

Rewrite $361$ as $19^{2}$ .

$f (\frac{7 + \sqrt{19}}{3}) = \frac{742 + 166 \sqrt{19}}{27} - 7 \frac{49 + 7 \sqrt{19} + 7 \sqrt{19} + \sqrt{19^{2}}}{9} + 10 (\frac{7 + \sqrt{19}}{3})$

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

$f (\frac{7 + \sqrt{19}}{3}) = \frac{742 + 166 \sqrt{19}}{27} - 7 \frac{49 + 7 \sqrt{19} + 7 \sqrt{19} + 19}{9} + 10 (\frac{7 + \sqrt{19}}{3})$

Add $49$ and $19$ .

$f (\frac{7 + \sqrt{19}}{3}) = \frac{742 + 166 \sqrt{19}}{27} - 7 \frac{68 + 7 \sqrt{19} + 7 \sqrt{19}}{9} + 10 (\frac{7 + \sqrt{19}}{3})$

Add $7 \sqrt{19}$ and $7 \sqrt{19}$ .

$f (\frac{7 + \sqrt{19}}{3}) = \frac{742 + 166 \sqrt{19}}{27} - 7 \frac{68 + 14 \sqrt{19}}{9} + 10 (\frac{7 + \sqrt{19}}{3})$

Combine $- 7$ and $\frac{68 + 14 \sqrt{19}}{9}$ .

$f (\frac{7 + \sqrt{19}}{3}) = \frac{742 + 166 \sqrt{19}}{27} + \frac{- 7 (68 + 14 \sqrt{19})}{9} + 10 (\frac{7 + \sqrt{19}}{3})$

Move the negative in front of the fraction.

$f (\frac{7 + \sqrt{19}}{3}) = \frac{742 + 166 \sqrt{19}}{27} - \frac{(7) (68 + 14 \sqrt{19})}{9} + 10 (\frac{7 + \sqrt{19}}{3})$

Combine $10$ and $\frac{7 + \sqrt{19}}{3}$ .

$f (\frac{7 + \sqrt{19}}{3}) = \frac{742 + 166 \sqrt{19}}{27} - \frac{7 (68 + 14 \sqrt{19})}{9} + \frac{10 (7 + \sqrt{19})}{3}$

Find the common denominator.

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Multiply $\frac{7 (68 + 14 \sqrt{19})}{9}$ by $\frac{3}{3}$ .

$f (\frac{7 + \sqrt{19}}{3}) = \frac{742 + 166 \sqrt{19}}{27} - (\frac{7 (68 + 14 \sqrt{19})}{9} \cdot \frac{3}{3}) + \frac{10 (7 + \sqrt{19})}{3}$

Multiply $\frac{7 (68 + 14 \sqrt{19})}{9}$ by $\frac{3}{3}$ .

$f (\frac{7 + \sqrt{19}}{3}) = \frac{742 + 166 \sqrt{19}}{27} - \frac{7 (68 + 14 \sqrt{19}) \cdot 3}{9 \cdot 3} + \frac{10 (7 + \sqrt{19})}{3}$

Multiply $\frac{10 (7 + \sqrt{19})}{3}$ by $\frac{9}{9}$ .

$f (\frac{7 + \sqrt{19}}{3}) = \frac{742 + 166 \sqrt{19}}{27} - \frac{7 (68 + 14 \sqrt{19}) \cdot 3}{9 \cdot 3} + \frac{10 (7 + \sqrt{19})}{3} \cdot \frac{9}{9}$

Multiply $\frac{10 (7 + \sqrt{19})}{3}$ by $\frac{9}{9}$ .

$f (\frac{7 + \sqrt{19}}{3}) = \frac{742 + 166 \sqrt{19}}{27} - \frac{7 (68 + 14 \sqrt{19}) \cdot 3}{9 \cdot 3} + \frac{10 (7 + \sqrt{19}) \cdot 9}{3 \cdot 9}$

Reorder the factors of $9 \cdot 3$ .

$f (\frac{7 + \sqrt{19}}{3}) = \frac{742 + 166 \sqrt{19}}{27} - \frac{7 (68 + 14 \sqrt{19}) \cdot 3}{3 \cdot 9} + \frac{10 (7 + \sqrt{19}) \cdot 9}{3 \cdot 9}$

Multiply $3$ by $9$ .

$f (\frac{7 + \sqrt{19}}{3}) = \frac{742 + 166 \sqrt{19}}{27} - \frac{7 (68 + 14 \sqrt{19}) \cdot 3}{27} + \frac{10 (7 + \sqrt{19}) \cdot 9}{3 \cdot 9}$

Multiply $3$ by $9$ .

$f (\frac{7 + \sqrt{19}}{3}) = \frac{742 + 166 \sqrt{19}}{27} - \frac{7 (68 + 14 \sqrt{19}) \cdot 3}{27} + \frac{10 (7 + \sqrt{19}) \cdot 9}{27}$

Combine the numerators over the common denominator.

$f (\frac{7 + \sqrt{19}}{3}) = \frac{742 + 166 \sqrt{19} - 7 (68 + 14 \sqrt{19}) \cdot 3 + 10 (7 + \sqrt{19}) \cdot 9}{27}$

Simplify each term.

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Apply the distributive property.

$f (\frac{7 + \sqrt{19}}{3}) = \frac{742 + 166 \sqrt{19} + (- 7 \cdot 68 - 7 (14 \sqrt{19})) \cdot 3 + 10 (7 + \sqrt{19}) \cdot 9}{27}$

Multiply $- 7$ by $68$ .

$f (\frac{7 + \sqrt{19}}{3}) = \frac{742 + 166 \sqrt{19} + (- 476 - 7 (14 \sqrt{19})) \cdot 3 + 10 (7 + \sqrt{19}) \cdot 9}{27}$

Multiply $14$ by $- 7$ .

$f (\frac{7 + \sqrt{19}}{3}) = \frac{742 + 166 \sqrt{19} + (- 476 - 98 \sqrt{19}) \cdot 3 + 10 (7 + \sqrt{19}) \cdot 9}{27}$

Apply the distributive property.

$f (\frac{7 + \sqrt{19}}{3}) = \frac{742 + 166 \sqrt{19} - 476 \cdot 3 - 98 \sqrt{19} \cdot 3 + 10 (7 + \sqrt{19}) \cdot 9}{27}$

Multiply $- 476$ by $3$ .

$f (\frac{7 + \sqrt{19}}{3}) = \frac{742 + 166 \sqrt{19} - 1428 - 98 \sqrt{19} \cdot 3 + 10 (7 + \sqrt{19}) \cdot 9}{27}$

Multiply $3$ by $- 98$ .

$f (\frac{7 + \sqrt{19}}{3}) = \frac{742 + 166 \sqrt{19} - 1428 - 294 \sqrt{19} + 10 (7 + \sqrt{19}) \cdot 9}{27}$

Apply the distributive property.

$f (\frac{7 + \sqrt{19}}{3}) = \frac{742 + 166 \sqrt{19} - 1428 - 294 \sqrt{19} + (10 \cdot 7 + 10 \sqrt{19}) \cdot 9}{27}$

Multiply $10$ by $7$ .

$f (\frac{7 + \sqrt{19}}{3}) = \frac{742 + 166 \sqrt{19} - 1428 - 294 \sqrt{19} + (70 + 10 \sqrt{19}) \cdot 9}{27}$

Apply the distributive property.

$f (\frac{7 + \sqrt{19}}{3}) = \frac{742 + 166 \sqrt{19} - 1428 - 294 \sqrt{19} + 70 \cdot 9 + 10 \sqrt{19} \cdot 9}{27}$

Multiply $70$ by $9$ .

$f (\frac{7 + \sqrt{19}}{3}) = \frac{742 + 166 \sqrt{19} - 1428 - 294 \sqrt{19} + 630 + 10 \sqrt{19} \cdot 9}{27}$

Multiply $9$ by $10$ .

$f (\frac{7 + \sqrt{19}}{3}) = \frac{742 + 166 \sqrt{19} - 1428 - 294 \sqrt{19} + 630 + 90 \sqrt{19}}{27}$

Simplify terms.

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Subtract $1428$ from $742$ .

$f (\frac{7 + \sqrt{19}}{3}) = \frac{- 686 + 166 \sqrt{19} - 294 \sqrt{19} + 630 + 90 \sqrt{19}}{27}$

Add $- 686$ and $630$ .

$f (\frac{7 + \sqrt{19}}{3}) = \frac{- 56 + 166 \sqrt{19} - 294 \sqrt{19} + 90 \sqrt{19}}{27}$

Subtract $294 \sqrt{19}$ from $166 \sqrt{19}$ .

$f (\frac{7 + \sqrt{19}}{3}) = \frac{- 56 - 128 \sqrt{19} + 90 \sqrt{19}}{27}$

Add $- 128 \sqrt{19}$ and $90 \sqrt{19}$ .

$f (\frac{7 + \sqrt{19}}{3}) = \frac{- 56 - 38 \sqrt{19}}{27}$

Rewrite $- 56$ as $- 1 (56)$ .

$f (\frac{7 + \sqrt{19}}{3}) = \frac{- 1 \cdot 56 - 38 \sqrt{19}}{27}$

Factor $- 1$ out of $- 38 \sqrt{19}$ .

$f (\frac{7 + \sqrt{19}}{3}) = \frac{- 1 \cdot 56 - (38 \sqrt{19})}{27}$

Factor $- 1$ out of $- 1 (56) - (38 \sqrt{19})$ .

$f (\frac{7 + \sqrt{19}}{3}) = \frac{- 1 (56 + 38 \sqrt{19})}{27}$

Move the negative in front of the fraction.

$f (\frac{7 + \sqrt{19}}{3}) = - \frac{56 + 38 \sqrt{19}}{27}$

The final answer is $- \frac{56 + 38 \sqrt{19}}{27}$ .

$y = - \frac{56 + 38 \sqrt{19}}{27}$

Step 13

Evaluate the second derivative at $x = \frac{7 - \sqrt{19}}{3}$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$6 (\frac{7 - \sqrt{19}}{3}) - 14$

Step 14

Evaluate the second derivative.

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Simplify each term.

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Cancel the common factor of $3$ .

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Factor $3$ out of $6$ .

$3 (2) \frac{7 - \sqrt{19}}{3} - 14$

Cancel the common factor.

$3 \cdot 2 \frac{7 - \sqrt{19}}{3} - 14$

Rewrite the expression.

$2 (7 - \sqrt{19}) - 14$

Apply the distributive property.

$2 \cdot 7 + 2 (- \sqrt{19}) - 14$

Multiply $2$ by $7$ .

$14 + 2 (- \sqrt{19}) - 14$

Multiply $- 1$ by $2$ .

$14 - 2 \sqrt{19} - 14$

Simplify by subtracting numbers.

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Subtract $14$ from $14$ .

$0 - 2 \sqrt{19}$

Subtract $2 \sqrt{19}$ from $0$ .

$- 2 \sqrt{19}$

Step 15

$x = \frac{7 - \sqrt{19}}{3}$ is a local maximum because the value of the second derivative is negative. This is referred to as the second derivative test.

$x = \frac{7 - \sqrt{19}}{3}$ is a local maximum

Step 16

Find the y-value when $x = \frac{7 - \sqrt{19}}{3}$ .

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Replace the variable $x$ with $\frac{7 - \sqrt{19}}{3}$ in the expression.

$f (\frac{7 - \sqrt{19}}{3}) = {(\frac{7 - \sqrt{19}}{3})}^{3} - 7 {(\frac{7 - \sqrt{19}}{3})}^{2} + 10 (\frac{7 - \sqrt{19}}{3})$

Simplify the result.

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Simplify each term.

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Apply the product rule to $\frac{7 - \sqrt{19}}{3}$ .

$f (\frac{7 - \sqrt{19}}{3}) = \frac{{(7 - \sqrt{19})}^{3}}{3^{3}} - 7 {(\frac{7 - \sqrt{19}}{3})}^{2} + 10 (\frac{7 - \sqrt{19}}{3})$

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

$f (\frac{7 - \sqrt{19}}{3}) = \frac{{(7 - \sqrt{19})}^{3}}{27} - 7 {(\frac{7 - \sqrt{19}}{3})}^{2} + 10 (\frac{7 - \sqrt{19}}{3})$

Use the Binomial Theorem.

$f (\frac{7 - \sqrt{19}}{3}) = \frac{7^{3} + 3 \cdot (7^{2} (- \sqrt{19})) + 3 \cdot (7 {(- \sqrt{19})}^{2}) + {(- \sqrt{19})}^{3}}{27} - 7 {(\frac{7 - \sqrt{19}}{3})}^{2} + 10 (\frac{7 - \sqrt{19}}{3})$

Simplify each term.

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Raise $7$ to the power of $3$ .

$f (\frac{7 - \sqrt{19}}{3}) = \frac{343 + 3 \cdot (7^{2} (- \sqrt{19})) + 3 \cdot (7 {(- \sqrt{19})}^{2}) + {(- \sqrt{19})}^{3}}{27} - 7 {(\frac{7 - \sqrt{19}}{3})}^{2} + 10 (\frac{7 - \sqrt{19}}{3})$

Raise $7$ to the power of $2$ .

$f (\frac{7 - \sqrt{19}}{3}) = \frac{343 + 3 \cdot (49 (- \sqrt{19})) + 3 \cdot (7 {(- \sqrt{19})}^{2}) + {(- \sqrt{19})}^{3}}{27} - 7 {(\frac{7 - \sqrt{19}}{3})}^{2} + 10 (\frac{7 - \sqrt{19}}{3})$

Multiply $3$ by $49$ .

$f (\frac{7 - \sqrt{19}}{3}) = \frac{343 + 147 (- \sqrt{19}) + 3 \cdot (7 {(- \sqrt{19})}^{2}) + {(- \sqrt{19})}^{3}}{27} - 7 {(\frac{7 - \sqrt{19}}{3})}^{2} + 10 (\frac{7 - \sqrt{19}}{3})$

Multiply $- 1$ by $147$ .

$f (\frac{7 - \sqrt{19}}{3}) = \frac{343 - 147 \sqrt{19} + 3 \cdot (7 {(- \sqrt{19})}^{2}) + {(- \sqrt{19})}^{3}}{27} - 7 {(\frac{7 - \sqrt{19}}{3})}^{2} + 10 (\frac{7 - \sqrt{19}}{3})$

Multiply $3$ by $7$ .

$f (\frac{7 - \sqrt{19}}{3}) = \frac{343 - 147 \sqrt{19} + 21 {(- \sqrt{19})}^{2} + {(- \sqrt{19})}^{3}}{27} - 7 {(\frac{7 - \sqrt{19}}{3})}^{2} + 10 (\frac{7 - \sqrt{19}}{3})$

Apply the product rule to $- \sqrt{19}$ .

$f (\frac{7 - \sqrt{19}}{3}) = \frac{343 - 147 \sqrt{19} + 21 ({(- 1)}^{2} {\sqrt{19}}^{2}) + {(- \sqrt{19})}^{3}}{27} - 7 {(\frac{7 - \sqrt{19}}{3})}^{2} + 10 (\frac{7 - \sqrt{19}}{3})$

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

$f (\frac{7 - \sqrt{19}}{3}) = \frac{343 - 147 \sqrt{19} + 21 (1 {\sqrt{19}}^{2}) + {(- \sqrt{19})}^{3}}{27} - 7 {(\frac{7 - \sqrt{19}}{3})}^{2} + 10 (\frac{7 - \sqrt{19}}{3})$

Multiply ${\sqrt{19}}^{2}$ by $1$ .

$f (\frac{7 - \sqrt{19}}{3}) = \frac{343 - 147 \sqrt{19} + 21 {\sqrt{19}}^{2} + {(- \sqrt{19})}^{3}}{27} - 7 {(\frac{7 - \sqrt{19}}{3})}^{2} + 10 (\frac{7 - \sqrt{19}}{3})$

Rewrite ${\sqrt{19}}^{2}$ as $19$ .

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Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{19}$ as $19^{\frac{1}{2}}$ .

$f (\frac{7 - \sqrt{19}}{3}) = \frac{343 - 147 \sqrt{19} + 21 {(19^{\frac{1}{2}})}^{2} + {(- \sqrt{19})}^{3}}{27} - 7 {(\frac{7 - \sqrt{19}}{3})}^{2} + 10 (\frac{7 - \sqrt{19}}{3})$

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

$f (\frac{7 - \sqrt{19}}{3}) = \frac{343 - 147 \sqrt{19} + 21 \cdot 19^{\frac{1}{2} \cdot 2} + {(- \sqrt{19})}^{3}}{27} - 7 {(\frac{7 - \sqrt{19}}{3})}^{2} + 10 (\frac{7 - \sqrt{19}}{3})$

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

$f (\frac{7 - \sqrt{19}}{3}) = \frac{343 - 147 \sqrt{19} + 21 \cdot 19^{\frac{2}{2}} + {(- \sqrt{19})}^{3}}{27} - 7 {(\frac{7 - \sqrt{19}}{3})}^{2} + 10 (\frac{7 - \sqrt{19}}{3})$

Cancel the common factor of $2$ .

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Cancel the common factor.

$f (\frac{7 - \sqrt{19}}{3}) = \frac{343 - 147 \sqrt{19} + 21 \cdot 19^{\frac{2}{2}} + {(- \sqrt{19})}^{3}}{27} - 7 {(\frac{7 - \sqrt{19}}{3})}^{2} + 10 (\frac{7 - \sqrt{19}}{3})$

Rewrite the expression.

$f (\frac{7 - \sqrt{19}}{3}) = \frac{343 - 147 \sqrt{19} + 21 \cdot 19 + {(- \sqrt{19})}^{3}}{27} - 7 {(\frac{7 - \sqrt{19}}{3})}^{2} + 10 (\frac{7 - \sqrt{19}}{3})$

Evaluate the exponent.

$f (\frac{7 - \sqrt{19}}{3}) = \frac{343 - 147 \sqrt{19} + 21 \cdot 19 + {(- \sqrt{19})}^{3}}{27} - 7 {(\frac{7 - \sqrt{19}}{3})}^{2} + 10 (\frac{7 - \sqrt{19}}{3})$

Multiply $21$ by $19$ .

$f (\frac{7 - \sqrt{19}}{3}) = \frac{343 - 147 \sqrt{19} + 399 + {(- \sqrt{19})}^{3}}{27} - 7 {(\frac{7 - \sqrt{19}}{3})}^{2} + 10 (\frac{7 - \sqrt{19}}{3})$

Apply the product rule to $- \sqrt{19}$ .

$f (\frac{7 - \sqrt{19}}{3}) = \frac{343 - 147 \sqrt{19} + 399 + {(- 1)}^{3} {\sqrt{19}}^{3}}{27} - 7 {(\frac{7 - \sqrt{19}}{3})}^{2} + 10 (\frac{7 - \sqrt{19}}{3})$

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

$f (\frac{7 - \sqrt{19}}{3}) = \frac{343 - 147 \sqrt{19} + 399 - {\sqrt{19}}^{3}}{27} - 7 {(\frac{7 - \sqrt{19}}{3})}^{2} + 10 (\frac{7 - \sqrt{19}}{3})$

Rewrite ${\sqrt{19}}^{3}$ as ${(19^{3})}^{\frac{1}{2}}$ .

$f (\frac{7 - \sqrt{19}}{3}) = \frac{343 - 147 \sqrt{19} + 399 - \sqrt{19^{3}}}{27} - 7 {(\frac{7 - \sqrt{19}}{3})}^{2} + 10 (\frac{7 - \sqrt{19}}{3})$

Raise $19$ to the power of $3$ .

$f (\frac{7 - \sqrt{19}}{3}) = \frac{343 - 147 \sqrt{19} + 399 - \sqrt{6859}}{27} - 7 {(\frac{7 - \sqrt{19}}{3})}^{2} + 10 (\frac{7 - \sqrt{19}}{3})$

Rewrite $6859$ as $19^{2} \cdot 19$ .

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Factor $361$ out of $6859$ .

$f (\frac{7 - \sqrt{19}}{3}) = \frac{343 - 147 \sqrt{19} + 399 - \sqrt{361 (19)}}{27} - 7 {(\frac{7 - \sqrt{19}}{3})}^{2} + 10 (\frac{7 - \sqrt{19}}{3})$

Rewrite $361$ as $19^{2}$ .

$f (\frac{7 - \sqrt{19}}{3}) = \frac{343 - 147 \sqrt{19} + 399 - \sqrt{19^{2} \cdot 19}}{27} - 7 {(\frac{7 - \sqrt{19}}{3})}^{2} + 10 (\frac{7 - \sqrt{19}}{3})$

Pull terms out from under the radical.

$f (\frac{7 - \sqrt{19}}{3}) = \frac{343 - 147 \sqrt{19} + 399 - (19 \sqrt{19})}{27} - 7 {(\frac{7 - \sqrt{19}}{3})}^{2} + 10 (\frac{7 - \sqrt{19}}{3})$

Multiply $19$ by $- 1$ .

$f (\frac{7 - \sqrt{19}}{3}) = \frac{343 - 147 \sqrt{19} + 399 - 19 \sqrt{19}}{27} - 7 {(\frac{7 - \sqrt{19}}{3})}^{2} + 10 (\frac{7 - \sqrt{19}}{3})$

Add $343$ and $399$ .

$f (\frac{7 - \sqrt{19}}{3}) = \frac{742 - 147 \sqrt{19} - 19 \sqrt{19}}{27} - 7 {(\frac{7 - \sqrt{19}}{3})}^{2} + 10 (\frac{7 - \sqrt{19}}{3})$

Subtract $19 \sqrt{19}$ from $- 147 \sqrt{19}$ .

$f (\frac{7 - \sqrt{19}}{3}) = \frac{742 - 166 \sqrt{19}}{27} - 7 {(\frac{7 - \sqrt{19}}{3})}^{2} + 10 (\frac{7 - \sqrt{19}}{3})$

Apply the product rule to $\frac{7 - \sqrt{19}}{3}$ .

$f (\frac{7 - \sqrt{19}}{3}) = \frac{742 - 166 \sqrt{19}}{27} - 7 \frac{{(7 - \sqrt{19})}^{2}}{3^{2}} + 10 (\frac{7 - \sqrt{19}}{3})$

Raise $3$ to the power of $2$ .

$f (\frac{7 - \sqrt{19}}{3}) = \frac{742 - 166 \sqrt{19}}{27} - 7 \frac{{(7 - \sqrt{19})}^{2}}{9} + 10 (\frac{7 - \sqrt{19}}{3})$

Rewrite ${(7 - \sqrt{19})}^{2}$ as $(7 - \sqrt{19}) (7 - \sqrt{19})$ .

$f (\frac{7 - \sqrt{19}}{3}) = \frac{742 - 166 \sqrt{19}}{27} - 7 \frac{(7 - \sqrt{19}) (7 - \sqrt{19})}{9} + 10 (\frac{7 - \sqrt{19}}{3})$

Expand $(7 - \sqrt{19}) (7 - \sqrt{19})$ using the FOIL Method.

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Apply the distributive property.

$f (\frac{7 - \sqrt{19}}{3}) = \frac{742 - 166 \sqrt{19}}{27} - 7 \frac{7 (7 - \sqrt{19}) - \sqrt{19} (7 - \sqrt{19})}{9} + 10 (\frac{7 - \sqrt{19}}{3})$

Apply the distributive property.

$f (\frac{7 - \sqrt{19}}{3}) = \frac{742 - 166 \sqrt{19}}{27} - 7 \frac{7 \cdot 7 + 7 (- \sqrt{19}) - \sqrt{19} (7 - \sqrt{19})}{9} + 10 (\frac{7 - \sqrt{19}}{3})$

Apply the distributive property.

$f (\frac{7 - \sqrt{19}}{3}) = \frac{742 - 166 \sqrt{19}}{27} - 7 \frac{7 \cdot 7 + 7 (- \sqrt{19}) - \sqrt{19} \cdot 7 - \sqrt{19} (- \sqrt{19})}{9} + 10 (\frac{7 - \sqrt{19}}{3})$

Simplify and combine like terms.

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Simplify each term.

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Multiply $7$ by $7$ .

$f (\frac{7 - \sqrt{19}}{3}) = \frac{742 - 166 \sqrt{19}}{27} - 7 \frac{49 + 7 (- \sqrt{19}) - \sqrt{19} \cdot 7 - \sqrt{19} (- \sqrt{19})}{9} + 10 (\frac{7 - \sqrt{19}}{3})$

Multiply $- 1$ by $7$ .

$f (\frac{7 - \sqrt{19}}{3}) = \frac{742 - 166 \sqrt{19}}{27} - 7 \frac{49 - 7 \sqrt{19} - \sqrt{19} \cdot 7 - \sqrt{19} (- \sqrt{19})}{9} + 10 (\frac{7 - \sqrt{19}}{3})$

Multiply $7$ by $- 1$ .

$f (\frac{7 - \sqrt{19}}{3}) = \frac{742 - 166 \sqrt{19}}{27} - 7 \frac{49 - 7 \sqrt{19} - 7 \sqrt{19} - \sqrt{19} (- \sqrt{19})}{9} + 10 (\frac{7 - \sqrt{19}}{3})$

Multiply $- \sqrt{19} (- \sqrt{19})$ .

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Multiply $- 1$ by $- 1$ .

$f (\frac{7 - \sqrt{19}}{3}) = \frac{742 - 166 \sqrt{19}}{27} - 7 \frac{49 - 7 \sqrt{19} - 7 \sqrt{19} + 1 \sqrt{19} \sqrt{19}}{9} + 10 (\frac{7 - \sqrt{19}}{3})$

Multiply $\sqrt{19}$ by $1$ .

$f (\frac{7 - \sqrt{19}}{3}) = \frac{742 - 166 \sqrt{19}}{27} - 7 \frac{49 - 7 \sqrt{19} - 7 \sqrt{19} + \sqrt{19} \sqrt{19}}{9} + 10 (\frac{7 - \sqrt{19}}{3})$

Raise $\sqrt{19}$ to the power of $1$ .

$f (\frac{7 - \sqrt{19}}{3}) = \frac{742 - 166 \sqrt{19}}{27} - 7 \frac{49 - 7 \sqrt{19} - 7 \sqrt{19} + \sqrt{19} \sqrt{19}}{9} + 10 (\frac{7 - \sqrt{19}}{3})$

Raise $\sqrt{19}$ to the power of $1$ .

$f (\frac{7 - \sqrt{19}}{3}) = \frac{742 - 166 \sqrt{19}}{27} - 7 \frac{49 - 7 \sqrt{19} - 7 \sqrt{19} + \sqrt{19} \sqrt{19}}{9} + 10 (\frac{7 - \sqrt{19}}{3})$

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

$f (\frac{7 - \sqrt{19}}{3}) = \frac{742 - 166 \sqrt{19}}{27} - 7 \frac{49 - 7 \sqrt{19} - 7 \sqrt{19} + {\sqrt{19}}^{1 + 1}}{9} + 10 (\frac{7 - \sqrt{19}}{3})$

Add $1$ and $1$ .

$f (\frac{7 - \sqrt{19}}{3}) = \frac{742 - 166 \sqrt{19}}{27} - 7 \frac{49 - 7 \sqrt{19} - 7 \sqrt{19} + {\sqrt{19}}^{2}}{9} + 10 (\frac{7 - \sqrt{19}}{3})$

Rewrite ${\sqrt{19}}^{2}$ as $19$ .

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Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{19}$ as $19^{\frac{1}{2}}$ .

$f (\frac{7 - \sqrt{19}}{3}) = \frac{742 - 166 \sqrt{19}}{27} - 7 \frac{49 - 7 \sqrt{19} - 7 \sqrt{19} + {(19^{\frac{1}{2}})}^{2}}{9} + 10 (\frac{7 - \sqrt{19}}{3})$

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

$f (\frac{7 - \sqrt{19}}{3}) = \frac{742 - 166 \sqrt{19}}{27} - 7 \frac{49 - 7 \sqrt{19} - 7 \sqrt{19} + 19^{\frac{1}{2} \cdot 2}}{9} + 10 (\frac{7 - \sqrt{19}}{3})$

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

$f (\frac{7 - \sqrt{19}}{3}) = \frac{742 - 166 \sqrt{19}}{27} - 7 \frac{49 - 7 \sqrt{19} - 7 \sqrt{19} + 19^{\frac{2}{2}}}{9} + 10 (\frac{7 - \sqrt{19}}{3})$

Cancel the common factor of $2$ .

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Cancel the common factor.

$f (\frac{7 - \sqrt{19}}{3}) = \frac{742 - 166 \sqrt{19}}{27} - 7 \frac{49 - 7 \sqrt{19} - 7 \sqrt{19} + 19^{\frac{2}{2}}}{9} + 10 (\frac{7 - \sqrt{19}}{3})$

Rewrite the expression.

$f (\frac{7 - \sqrt{19}}{3}) = \frac{742 - 166 \sqrt{19}}{27} - 7 \frac{49 - 7 \sqrt{19} - 7 \sqrt{19} + 19}{9} + 10 (\frac{7 - \sqrt{19}}{3})$

Evaluate the exponent.

$f (\frac{7 - \sqrt{19}}{3}) = \frac{742 - 166 \sqrt{19}}{27} - 7 \frac{49 - 7 \sqrt{19} - 7 \sqrt{19} + 19}{9} + 10 (\frac{7 - \sqrt{19}}{3})$

Add $49$ and $19$ .

$f (\frac{7 - \sqrt{19}}{3}) = \frac{742 - 166 \sqrt{19}}{27} - 7 \frac{68 - 7 \sqrt{19} - 7 \sqrt{19}}{9} + 10 (\frac{7 - \sqrt{19}}{3})$

Subtract $7 \sqrt{19}$ from $- 7 \sqrt{19}$ .

$f (\frac{7 - \sqrt{19}}{3}) = \frac{742 - 166 \sqrt{19}}{27} - 7 \frac{68 - 14 \sqrt{19}}{9} + 10 (\frac{7 - \sqrt{19}}{3})$

Combine $- 7$ and $\frac{68 - 14 \sqrt{19}}{9}$ .

$f (\frac{7 - \sqrt{19}}{3}) = \frac{742 - 166 \sqrt{19}}{27} + \frac{- 7 (68 - 14 \sqrt{19})}{9} + 10 (\frac{7 - \sqrt{19}}{3})$

Move the negative in front of the fraction.

$f (\frac{7 - \sqrt{19}}{3}) = \frac{742 - 166 \sqrt{19}}{27} - \frac{(7) (68 - 14 \sqrt{19})}{9} + 10 (\frac{7 - \sqrt{19}}{3})$

Combine $10$ and $\frac{7 - \sqrt{19}}{3}$ .

$f (\frac{7 - \sqrt{19}}{3}) = \frac{742 - 166 \sqrt{19}}{27} - \frac{7 (68 - 14 \sqrt{19})}{9} + \frac{10 (7 - \sqrt{19})}{3}$

Find the common denominator.

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Multiply $\frac{7 (68 - 14 \sqrt{19})}{9}$ by $\frac{3}{3}$ .

$f (\frac{7 - \sqrt{19}}{3}) = \frac{742 - 166 \sqrt{19}}{27} - (\frac{7 (68 - 14 \sqrt{19})}{9} \cdot \frac{3}{3}) + \frac{10 (7 - \sqrt{19})}{3}$

Multiply $\frac{7 (68 - 14 \sqrt{19})}{9}$ by $\frac{3}{3}$ .

$f (\frac{7 - \sqrt{19}}{3}) = \frac{742 - 166 \sqrt{19}}{27} - \frac{7 (68 - 14 \sqrt{19}) \cdot 3}{9 \cdot 3} + \frac{10 (7 - \sqrt{19})}{3}$

Multiply $\frac{10 (7 - \sqrt{19})}{3}$ by $\frac{9}{9}$ .

$f (\frac{7 - \sqrt{19}}{3}) = \frac{742 - 166 \sqrt{19}}{27} - \frac{7 (68 - 14 \sqrt{19}) \cdot 3}{9 \cdot 3} + \frac{10 (7 - \sqrt{19})}{3} \cdot \frac{9}{9}$

Multiply $\frac{10 (7 - \sqrt{19})}{3}$ by $\frac{9}{9}$ .

$f (\frac{7 - \sqrt{19}}{3}) = \frac{742 - 166 \sqrt{19}}{27} - \frac{7 (68 - 14 \sqrt{19}) \cdot 3}{9 \cdot 3} + \frac{10 (7 - \sqrt{19}) \cdot 9}{3 \cdot 9}$

Reorder the factors of $9 \cdot 3$ .

$f (\frac{7 - \sqrt{19}}{3}) = \frac{742 - 166 \sqrt{19}}{27} - \frac{7 (68 - 14 \sqrt{19}) \cdot 3}{3 \cdot 9} + \frac{10 (7 - \sqrt{19}) \cdot 9}{3 \cdot 9}$

Multiply $3$ by $9$ .

$f (\frac{7 - \sqrt{19}}{3}) = \frac{742 - 166 \sqrt{19}}{27} - \frac{7 (68 - 14 \sqrt{19}) \cdot 3}{27} + \frac{10 (7 - \sqrt{19}) \cdot 9}{3 \cdot 9}$

Multiply $3$ by $9$ .

$f (\frac{7 - \sqrt{19}}{3}) = \frac{742 - 166 \sqrt{19}}{27} - \frac{7 (68 - 14 \sqrt{19}) \cdot 3}{27} + \frac{10 (7 - \sqrt{19}) \cdot 9}{27}$

Combine the numerators over the common denominator.

$f (\frac{7 - \sqrt{19}}{3}) = \frac{742 - 166 \sqrt{19} - 7 (68 - 14 \sqrt{19}) \cdot 3 + 10 (7 - \sqrt{19}) \cdot 9}{27}$

Simplify each term.

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Apply the distributive property.

$f (\frac{7 - \sqrt{19}}{3}) = \frac{742 - 166 \sqrt{19} + (- 7 \cdot 68 - 7 (- 14 \sqrt{19})) \cdot 3 + 10 (7 - \sqrt{19}) \cdot 9}{27}$

Multiply $- 7$ by $68$ .

$f (\frac{7 - \sqrt{19}}{3}) = \frac{742 - 166 \sqrt{19} + (- 476 - 7 (- 14 \sqrt{19})) \cdot 3 + 10 (7 - \sqrt{19}) \cdot 9}{27}$

Multiply $- 14$ by $- 7$ .

$f (\frac{7 - \sqrt{19}}{3}) = \frac{742 - 166 \sqrt{19} + (- 476 + 98 \sqrt{19}) \cdot 3 + 10 (7 - \sqrt{19}) \cdot 9}{27}$

Apply the distributive property.

$f (\frac{7 - \sqrt{19}}{3}) = \frac{742 - 166 \sqrt{19} - 476 \cdot 3 + 98 \sqrt{19} \cdot 3 + 10 (7 - \sqrt{19}) \cdot 9}{27}$

Multiply $- 476$ by $3$ .

$f (\frac{7 - \sqrt{19}}{3}) = \frac{742 - 166 \sqrt{19} - 1428 + 98 \sqrt{19} \cdot 3 + 10 (7 - \sqrt{19}) \cdot 9}{27}$

Multiply $3$ by $98$ .

$f (\frac{7 - \sqrt{19}}{3}) = \frac{742 - 166 \sqrt{19} - 1428 + 294 \sqrt{19} + 10 (7 - \sqrt{19}) \cdot 9}{27}$

Apply the distributive property.

$f (\frac{7 - \sqrt{19}}{3}) = \frac{742 - 166 \sqrt{19} - 1428 + 294 \sqrt{19} + (10 \cdot 7 + 10 (- \sqrt{19})) \cdot 9}{27}$

Multiply $10$ by $7$ .

$f (\frac{7 - \sqrt{19}}{3}) = \frac{742 - 166 \sqrt{19} - 1428 + 294 \sqrt{19} + (70 + 10 (- \sqrt{19})) \cdot 9}{27}$

Multiply $- 1$ by $10$ .

$f (\frac{7 - \sqrt{19}}{3}) = \frac{742 - 166 \sqrt{19} - 1428 + 294 \sqrt{19} + (70 - 10 \sqrt{19}) \cdot 9}{27}$

Apply the distributive property.

$f (\frac{7 - \sqrt{19}}{3}) = \frac{742 - 166 \sqrt{19} - 1428 + 294 \sqrt{19} + 70 \cdot 9 - 10 \sqrt{19} \cdot 9}{27}$

Multiply $70$ by $9$ .

$f (\frac{7 - \sqrt{19}}{3}) = \frac{742 - 166 \sqrt{19} - 1428 + 294 \sqrt{19} + 630 - 10 \sqrt{19} \cdot 9}{27}$

Multiply $9$ by $- 10$ .

$f (\frac{7 - \sqrt{19}}{3}) = \frac{742 - 166 \sqrt{19} - 1428 + 294 \sqrt{19} + 630 - 90 \sqrt{19}}{27}$

Simplify terms.

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Subtract $1428$ from $742$ .

$f (\frac{7 - \sqrt{19}}{3}) = \frac{- 686 - 166 \sqrt{19} + 294 \sqrt{19} + 630 - 90 \sqrt{19}}{27}$

Add $- 686$ and $630$ .

$f (\frac{7 - \sqrt{19}}{3}) = \frac{- 56 - 166 \sqrt{19} + 294 \sqrt{19} - 90 \sqrt{19}}{27}$

Add $- 166 \sqrt{19}$ and $294 \sqrt{19}$ .

$f (\frac{7 - \sqrt{19}}{3}) = \frac{- 56 + 128 \sqrt{19} - 90 \sqrt{19}}{27}$

Subtract $90 \sqrt{19}$ from $128 \sqrt{19}$ .

$f (\frac{7 - \sqrt{19}}{3}) = \frac{- 56 + 38 \sqrt{19}}{27}$

Rewrite $- 56$ as $- 1 (56)$ .

$f (\frac{7 - \sqrt{19}}{3}) = \frac{- 1 \cdot 56 + 38 \sqrt{19}}{27}$

Factor $- 1$ out of $38 \sqrt{19}$ .

$f (\frac{7 - \sqrt{19}}{3}) = \frac{- 1 \cdot 56 - (- 38 \sqrt{19})}{27}$

Factor $- 1$ out of $- 1 (56) - (- 38 \sqrt{19})$ .

$f (\frac{7 - \sqrt{19}}{3}) = \frac{- 1 (56 - 38 \sqrt{19})}{27}$

Move the negative in front of the fraction.

$f (\frac{7 - \sqrt{19}}{3}) = - \frac{56 - 38 \sqrt{19}}{27}$

The final answer is $- \frac{56 - 38 \sqrt{19}}{27}$ .

$y = - \frac{56 - 38 \sqrt{19}}{27}$

Step 17

These are the local extrema for $f (x) = x^{3} - 7 x^{2} + 10 x$ .

$(\frac{7 + \sqrt{19}}{3}, - \frac{56 + 38 \sqrt{19}}{27})$ is a local minima

$(\frac{7 - \sqrt{19}}{3}, - \frac{56 - 38 \sqrt{19}}{27})$ is a local maxima

Step 18