Calculus Examples

$p (x) = (x - 7) (x + 4) (x - 2)$

Step 1

Find the first derivative.

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Step 1.1

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = (x - 7) (x + 4)$ and $g (x) = x - 2$ .

$(x - 7) (x + 4) \frac{d}{d x} [x - 2] + (x - 2) \frac{d}{d x} [(x - 7) (x + 4)]$

Step 1.2

Differentiate.

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Step 1.2.1

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

$(x - 7) (x + 4) (\frac{d}{d x} [x] + \frac{d}{d x} [- 2]) + (x - 2) \frac{d}{d x} [(x - 7) (x + 4)]$

Step 1.2.2

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

$(x - 7) (x + 4) (1 + \frac{d}{d x} [- 2]) + (x - 2) \frac{d}{d x} [(x - 7) (x + 4)]$

Step 1.2.3

Since $- 2$ is constant with respect to $x$ , the derivative of $- 2$ with respect to $x$ is $0$ .

$(x - 7) (x + 4) (1 + 0) + (x - 2) \frac{d}{d x} [(x - 7) (x + 4)]$

Step 1.2.4

Simplify the expression.

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Step 1.2.4.1

Add $1$ and $0$ .

$(x - 7) (x + 4) \cdot 1 + (x - 2) \frac{d}{d x} [(x - 7) (x + 4)]$

Step 1.2.4.2

Multiply $x - 7$ by $1$ .

$(x - 7) (x + 4) + (x - 2) \frac{d}{d x} [(x - 7) (x + 4)]$

Step 1.3

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x - 7$ and $g (x) = x + 4$ .

$(x - 7) (x + 4) + (x - 2) ((x - 7) \frac{d}{d x} [x + 4] + (x + 4) \frac{d}{d x} [x - 7])$

Step 1.4

Differentiate.

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Step 1.4.1

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

$(x - 7) (x + 4) + (x - 2) ((x - 7) (\frac{d}{d x} [x] + \frac{d}{d x} [4]) + (x + 4) \frac{d}{d x} [x - 7])$

Step 1.4.2

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

$(x - 7) (x + 4) + (x - 2) ((x - 7) (1 + \frac{d}{d x} [4]) + (x + 4) \frac{d}{d x} [x - 7])$

Step 1.4.3

Since $4$ is constant with respect to $x$ , the derivative of $4$ with respect to $x$ is $0$ .

$(x - 7) (x + 4) + (x - 2) ((x - 7) (1 + 0) + (x + 4) \frac{d}{d x} [x - 7])$

Step 1.4.4

Simplify the expression.

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Step 1.4.4.1

Add $1$ and $0$ .

$(x - 7) (x + 4) + (x - 2) ((x - 7) \cdot 1 + (x + 4) \frac{d}{d x} [x - 7])$

Step 1.4.4.2

Multiply $x - 7$ by $1$ .

$(x - 7) (x + 4) + (x - 2) (x - 7 + (x + 4) \frac{d}{d x} [x - 7])$

Step 1.4.5

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

$(x - 7) (x + 4) + (x - 2) (x - 7 + (x + 4) (\frac{d}{d x} [x] + \frac{d}{d x} [- 7]))$

Step 1.4.6

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

$(x - 7) (x + 4) + (x - 2) (x - 7 + (x + 4) (1 + \frac{d}{d x} [- 7]))$

Step 1.4.7

Since $- 7$ is constant with respect to $x$ , the derivative of $- 7$ with respect to $x$ is $0$ .

$(x - 7) (x + 4) + (x - 2) (x - 7 + (x + 4) (1 + 0))$

Step 1.4.8

Simplify by adding terms.

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Step 1.4.8.1

Add $1$ and $0$ .

$(x - 7) (x + 4) + (x - 2) (x - 7 + (x + 4) \cdot 1)$

Step 1.4.8.2

Multiply $x + 4$ by $1$ .

$(x - 7) (x + 4) + (x - 2) (x - 7 + x + 4)$

Step 1.4.8.3

Add $x$ and $x$ .

$(x - 7) (x + 4) + (x - 2) (2 x - 7 + 4)$

Step 1.4.8.4

Add $- 7$ and $4$ .

$(x - 7) (x + 4) + (x - 2) (2 x - 3)$

Step 1.5

Simplify.

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Step 1.5.1

Apply the distributive property.

$(x - 7) x + (x - 7) \cdot 4 + (x - 2) (2 x - 3)$

Step 1.5.2

Apply the distributive property.

$x \cdot x - 7 x + (x - 7) \cdot 4 + (x - 2) (2 x - 3)$

Step 1.5.3

Apply the distributive property.

$x \cdot x - 7 x + x \cdot 4 - 7 \cdot 4 + (x - 2) (2 x - 3)$

Step 1.5.4

Apply the distributive property.

$x \cdot x - 7 x + x \cdot 4 - 7 \cdot 4 + (x - 2) (2 x) + (x - 2) \cdot - 3$

Step 1.5.5

Apply the distributive property.

$x \cdot x - 7 x + x \cdot 4 - 7 \cdot 4 + x (2 x) - 2 (2 x) + (x - 2) \cdot - 3$

Step 1.5.6

Apply the distributive property.

$x \cdot x - 7 x + x \cdot 4 - 7 \cdot 4 + x (2 x) - 2 (2 x) + x \cdot - 3 - 2 \cdot - 3$

Step 1.5.7

Combine terms.

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Step 1.5.7.1

Raise $x$ to the power of $1$ .

$x^{1} x - 7 x + x \cdot 4 - 7 \cdot 4 + x (2 x) - 2 (2 x) + x \cdot - 3 - 2 \cdot - 3$

Step 1.5.7.2

Raise $x$ to the power of $1$ .

$x^{1} x^{1} - 7 x + x \cdot 4 - 7 \cdot 4 + x (2 x) - 2 (2 x) + x \cdot - 3 - 2 \cdot - 3$

Step 1.5.7.3

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

$x^{1 + 1} - 7 x + x \cdot 4 - 7 \cdot 4 + x (2 x) - 2 (2 x) + x \cdot - 3 - 2 \cdot - 3$

Step 1.5.7.4

Add $1$ and $1$ .

$x^{2} - 7 x + x \cdot 4 - 7 \cdot 4 + x (2 x) - 2 (2 x) + x \cdot - 3 - 2 \cdot - 3$

Step 1.5.7.5

Move $4$ to the left of $x$ .

$x^{2} - 7 x + 4 \cdot x - 7 \cdot 4 + x (2 x) - 2 (2 x) + x \cdot - 3 - 2 \cdot - 3$

Step 1.5.7.6

Multiply $- 7$ by $4$ .

$x^{2} - 7 x + 4 x - 28 + x (2 x) - 2 (2 x) + x \cdot - 3 - 2 \cdot - 3$

Step 1.5.7.7

Add $- 7 x$ and $4 x$ .

$x^{2} - 3 x - 28 + x (2 x) - 2 (2 x) + x \cdot - 3 - 2 \cdot - 3$

Step 1.5.7.8

Raise $x$ to the power of $1$ .

$x^{2} - 3 x - 28 + 2 (x^{1} x) - 2 (2 x) + x \cdot - 3 - 2 \cdot - 3$

Step 1.5.7.9

Raise $x$ to the power of $1$ .

$x^{2} - 3 x - 28 + 2 (x^{1} x^{1}) - 2 (2 x) + x \cdot - 3 - 2 \cdot - 3$

Step 1.5.7.10

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

$x^{2} - 3 x - 28 + 2 x^{1 + 1} - 2 (2 x) + x \cdot - 3 - 2 \cdot - 3$

Step 1.5.7.11

Add $1$ and $1$ .

$x^{2} - 3 x - 28 + 2 x^{2} - 2 (2 x) + x \cdot - 3 - 2 \cdot - 3$

Step 1.5.7.12

Multiply $2$ by $- 2$ .

$x^{2} - 3 x - 28 + 2 x^{2} - 4 x + x \cdot - 3 - 2 \cdot - 3$

Step 1.5.7.13

Move $- 3$ to the left of $x$ .

$x^{2} - 3 x - 28 + 2 x^{2} - 4 x - 3 \cdot x - 2 \cdot - 3$

Step 1.5.7.14

Multiply $- 2$ by $- 3$ .

$x^{2} - 3 x - 28 + 2 x^{2} - 4 x - 3 x + 6$

Step 1.5.7.15

Subtract $3 x$ from $- 4 x$ .

$x^{2} - 3 x - 28 + 2 x^{2} - 7 x + 6$

Step 1.5.7.16

Add $x^{2}$ and $2 x^{2}$ .

$3 x^{2} - 3 x - 28 - 7 x + 6$

Step 1.5.7.17

Subtract $7 x$ from $- 3 x$ .

$3 x^{2} - 10 x - 28 + 6$

Step 1.5.7.18

Add $- 28$ and $6$ .

$3 x^{2} - 10 x - 22$

Step 2

Set the first derivative equal to $0$ and solve for $x$ .

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Step 2.1

Use the quadratic formula to find the solutions.

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

Step 2.2

Substitute the values $a = 3$ , $b = - 10$ , and $c = - 22$ into the quadratic formula and solve for $x$ .

$\frac{10 \pm \sqrt{{(- 10)}^{2} - 4 \cdot (3 \cdot - 22)}}{2 \cdot 3}$

Step 2.3

Simplify.

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Step 2.3.1

Simplify the numerator.

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Step 2.3.1.1

Raise $- 10$ to the power of $2$ .

$x = \frac{10 \pm \sqrt{100 - 4 \cdot 3 \cdot - 22}}{2 \cdot 3}$

Step 2.3.1.2

Multiply $- 4 \cdot 3 \cdot - 22$ .

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Step 2.3.1.2.1

Multiply $- 4$ by $3$ .

$x = \frac{10 \pm \sqrt{100 - 12 \cdot - 22}}{2 \cdot 3}$

Step 2.3.1.2.2

Multiply $- 12$ by $- 22$ .

$x = \frac{10 \pm \sqrt{100 + 264}}{2 \cdot 3}$

Step 2.3.1.3

Add $100$ and $264$ .

$x = \frac{10 \pm \sqrt{364}}{2 \cdot 3}$

Step 2.3.1.4

Rewrite $364$ as $2^{2} \cdot 91$ .

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Step 2.3.1.4.1

Factor $4$ out of $364$ .

$x = \frac{10 \pm \sqrt{4 (91)}}{2 \cdot 3}$

Step 2.3.1.4.2

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

$x = \frac{10 \pm \sqrt{2^{2} \cdot 91}}{2 \cdot 3}$

Step 2.3.1.5

Pull terms out from under the radical.

$x = \frac{10 \pm 2 \sqrt{91}}{2 \cdot 3}$

Step 2.3.2

Multiply $2$ by $3$ .

$x = \frac{10 \pm 2 \sqrt{91}}{6}$

Step 2.3.3

Simplify $\frac{10 \pm 2 \sqrt{91}}{6}$ .

$x = \frac{5 \pm \sqrt{91}}{3}$

Step 2.4

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

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Step 2.4.1

Simplify the numerator.

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Step 2.4.1.1

Raise $- 10$ to the power of $2$ .

$x = \frac{10 \pm \sqrt{100 - 4 \cdot 3 \cdot - 22}}{2 \cdot 3}$

Step 2.4.1.2

Multiply $- 4 \cdot 3 \cdot - 22$ .

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Step 2.4.1.2.1

Multiply $- 4$ by $3$ .

$x = \frac{10 \pm \sqrt{100 - 12 \cdot - 22}}{2 \cdot 3}$

Step 2.4.1.2.2

Multiply $- 12$ by $- 22$ .

$x = \frac{10 \pm \sqrt{100 + 264}}{2 \cdot 3}$

Step 2.4.1.3

Add $100$ and $264$ .

$x = \frac{10 \pm \sqrt{364}}{2 \cdot 3}$

Step 2.4.1.4

Rewrite $364$ as $2^{2} \cdot 91$ .

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Step 2.4.1.4.1

Factor $4$ out of $364$ .

$x = \frac{10 \pm \sqrt{4 (91)}}{2 \cdot 3}$

Step 2.4.1.4.2

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

$x = \frac{10 \pm \sqrt{2^{2} \cdot 91}}{2 \cdot 3}$

Step 2.4.1.5

Pull terms out from under the radical.

$x = \frac{10 \pm 2 \sqrt{91}}{2 \cdot 3}$

Step 2.4.2

Multiply $2$ by $3$ .

$x = \frac{10 \pm 2 \sqrt{91}}{6}$

Step 2.4.3

Simplify $\frac{10 \pm 2 \sqrt{91}}{6}$ .

$x = \frac{5 \pm \sqrt{91}}{3}$

Step 2.4.4

Change the $\pm$ to $+$ .

$x = \frac{5 + \sqrt{91}}{3}$

Step 2.5

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

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Step 2.5.1

Simplify the numerator.

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Step 2.5.1.1

Raise $- 10$ to the power of $2$ .

$x = \frac{10 \pm \sqrt{100 - 4 \cdot 3 \cdot - 22}}{2 \cdot 3}$

Step 2.5.1.2

Multiply $- 4 \cdot 3 \cdot - 22$ .

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Step 2.5.1.2.1

Multiply $- 4$ by $3$ .

$x = \frac{10 \pm \sqrt{100 - 12 \cdot - 22}}{2 \cdot 3}$

Step 2.5.1.2.2

Multiply $- 12$ by $- 22$ .

$x = \frac{10 \pm \sqrt{100 + 264}}{2 \cdot 3}$

Step 2.5.1.3

Add $100$ and $264$ .

$x = \frac{10 \pm \sqrt{364}}{2 \cdot 3}$

Step 2.5.1.4

Rewrite $364$ as $2^{2} \cdot 91$ .

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Step 2.5.1.4.1

Factor $4$ out of $364$ .

$x = \frac{10 \pm \sqrt{4 (91)}}{2 \cdot 3}$

Step 2.5.1.4.2

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

$x = \frac{10 \pm \sqrt{2^{2} \cdot 91}}{2 \cdot 3}$

Step 2.5.1.5

Pull terms out from under the radical.

$x = \frac{10 \pm 2 \sqrt{91}}{2 \cdot 3}$

Step 2.5.2

Multiply $2$ by $3$ .

$x = \frac{10 \pm 2 \sqrt{91}}{6}$

Step 2.5.3

Simplify $\frac{10 \pm 2 \sqrt{91}}{6}$ .

$x = \frac{5 \pm \sqrt{91}}{3}$

Step 2.5.4

Change the $\pm$ to $-$ .

$x = \frac{5 - \sqrt{91}}{3}$

Step 2.6

The final answer is the combination of both solutions.

$x = \frac{5 + \sqrt{91}}{3}, \frac{5 - \sqrt{91}}{3}$

Step 3

Split $(- \infty, \infty)$ into separate intervals around the $x$ values that make the first derivative $0$ or undefined.

$(- \infty, \frac{5 - \sqrt{91}}{3}) \cup (\frac{5 - \sqrt{91}}{3}, \frac{5 + \sqrt{91}}{3}) \cup (\frac{5 + \sqrt{91}}{3}, \infty)$

Step 4

Substitute any number, such as $- 4$ , from the interval $(- \infty, \frac{5 - \sqrt{91}}{3})$ in the first derivative $3 x^{2} - 10 x - 22$ to check if the result is negative or positive.

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Step 4.1

Replace the variable $x$ with $- 4$ in the expression.

$p' (- 4) = 3 {(- 4)}^{2} - 10 \cdot - 4 - 22$

Step 4.2

Simplify the result.

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Step 4.2.1

Simplify each term.

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Step 4.2.1.1

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

$p' (- 4) = 3 \cdot 16 - 10 \cdot - 4 - 22$

Step 4.2.1.2

Multiply $3$ by $16$ .

$p' (- 4) = 48 - 10 \cdot - 4 - 22$

Step 4.2.1.3

Multiply $- 10$ by $- 4$ .

$p' (- 4) = 48 + 40 - 22$

Step 4.2.2

Simplify by adding and subtracting.

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Step 4.2.2.1

Add $48$ and $40$ .

$p' (- 4) = 88 - 22$

Step 4.2.2.2

Subtract $22$ from $88$ .

$p' (- 4) = 66$

Step 4.2.3

The final answer is $66$ .

$66$

Step 5

Substitute any number, such as $0$ , from the interval $(\frac{5 - \sqrt{91}}{3}, \frac{5 + \sqrt{91}}{3})$ in the first derivative $3 x^{2} - 10 x - 22$ to check if the result is negative or positive.

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Step 5.1

Replace the variable $x$ with $0$ in the expression.

$p' (0) = 3 {(0)}^{2} - 10 \cdot 0 - 22$

Step 5.2

Simplify the result.

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Step 5.2.1

Simplify each term.

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Step 5.2.1.1

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

$p' (0) = 3 \cdot 0 - 10 \cdot 0 - 22$

Step 5.2.1.2

Multiply $3$ by $0$ .

$p' (0) = 0 - 10 \cdot 0 - 22$

Step 5.2.1.3

Multiply $- 10$ by $0$ .

$p' (0) = 0 + 0 - 22$

Step 5.2.2

Simplify by adding and subtracting.

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Step 5.2.2.1

Add $0$ and $0$ .

$p' (0) = 0 - 22$

Step 5.2.2.2

Subtract $22$ from $0$ .

$p' (0) = - 22$

Step 5.2.3

The final answer is $- 22$ .

$- 22$

Step 6

Substitute any number, such as $7$ , from the interval $(\frac{5 + \sqrt{91}}{3}, \infty)$ in the first derivative $3 x^{2} - 10 x - 22$ to check if the result is negative or positive.

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Step 6.1

Replace the variable $x$ with $7$ in the expression.

$p' (7) = 3 {(7)}^{2} - 10 \cdot 7 - 22$

Step 6.2

Simplify the result.

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Step 6.2.1

Simplify each term.

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Step 6.2.1.1

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

$p' (7) = 3 \cdot 49 - 10 \cdot 7 - 22$

Step 6.2.1.2

Multiply $3$ by $49$ .

$p' (7) = 147 - 10 \cdot 7 - 22$

Step 6.2.1.3

Multiply $- 10$ by $7$ .

$p' (7) = 147 - 70 - 22$

Step 6.2.2

Simplify by subtracting numbers.

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Step 6.2.2.1

Subtract $70$ from $147$ .

$p' (7) = 77 - 22$

Step 6.2.2.2

Subtract $22$ from $77$ .

$p' (7) = 55$

Step 6.2.3

The final answer is $55$ .

$55$

Step 7

Since the first derivative changed signs from positive to negative around $x = \frac{5 - \sqrt{91}}{3}$ , then there is a turning point at $x = \frac{5 - \sqrt{91}}{3}$ .

Step 8

Find the y-coordinate of $\frac{5 - \sqrt{91}}{3}$ to find the turning point.

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Step 8.1

Find $p (\frac{5 - \sqrt{91}}{3})$ to find the y-coordinate of $\frac{5 - \sqrt{91}}{3}$ .

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Step 8.1.1

Replace the variable $x$ with $\frac{5 - \sqrt{91}}{3}$ in the expression.

$p (\frac{5 - \sqrt{91}}{3}) = ((\frac{5 - \sqrt{91}}{3}) - 7) ((\frac{5 - \sqrt{91}}{3}) + 4) ((\frac{5 - \sqrt{91}}{3}) - 2)$

Step 8.1.2

Simplify $((\frac{5 - \sqrt{91}}{3}) - 7) ((\frac{5 - \sqrt{91}}{3}) + 4) ((\frac{5 - \sqrt{91}}{3}) - 2)$ .

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Step 8.1.2.1

Remove parentheses.

$((\frac{5 - \sqrt{91}}{3}) - 7) ((\frac{5 - \sqrt{91}}{3}) + 4) ((\frac{5 - \sqrt{91}}{3}) - 2)$

Step 8.1.2.2

To write $- 7$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$(\frac{5 - \sqrt{91}}{3} - 7 \cdot \frac{3}{3}) ((\frac{5 - \sqrt{91}}{3}) + 4) ((\frac{5 - \sqrt{91}}{3}) - 2)$

Step 8.1.2.3

Combine fractions.

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Step 8.1.2.3.1

Combine $- 7$ and $\frac{3}{3}$ .

$(\frac{5 - \sqrt{91}}{3} + \frac{- 7 \cdot 3}{3}) ((\frac{5 - \sqrt{91}}{3}) + 4) ((\frac{5 - \sqrt{91}}{3}) - 2)$

Step 8.1.2.3.2

Combine the numerators over the common denominator.

$\frac{5 - \sqrt{91} - 7 \cdot 3}{3} ((\frac{5 - \sqrt{91}}{3}) + 4) ((\frac{5 - \sqrt{91}}{3}) - 2)$

Step 8.1.2.4

Simplify the numerator.

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Step 8.1.2.4.1

Multiply $- 7$ by $3$ .

$\frac{5 - \sqrt{91} - 21}{3} ((\frac{5 - \sqrt{91}}{3}) + 4) ((\frac{5 - \sqrt{91}}{3}) - 2)$

Step 8.1.2.4.2

Subtract $21$ from $5$ .

$\frac{- 16 - \sqrt{91}}{3} ((\frac{5 - \sqrt{91}}{3}) + 4) ((\frac{5 - \sqrt{91}}{3}) - 2)$

Step 8.1.2.5

To write $4$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{- 16 - \sqrt{91}}{3} (\frac{5 - \sqrt{91}}{3} + 4 \cdot \frac{3}{3}) ((\frac{5 - \sqrt{91}}{3}) - 2)$

Step 8.1.2.6

Combine fractions.

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Step 8.1.2.6.1

Combine $4$ and $\frac{3}{3}$ .

$\frac{- 16 - \sqrt{91}}{3} (\frac{5 - \sqrt{91}}{3} + \frac{4 \cdot 3}{3}) ((\frac{5 - \sqrt{91}}{3}) - 2)$

Step 8.1.2.6.2

Combine the numerators over the common denominator.

$\frac{- 16 - \sqrt{91}}{3} \cdot \frac{5 - \sqrt{91} + 4 \cdot 3}{3} ((\frac{5 - \sqrt{91}}{3}) - 2)$

Step 8.1.2.7

Simplify the numerator.

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Step 8.1.2.7.1

Multiply $4$ by $3$ .

$\frac{- 16 - \sqrt{91}}{3} \cdot \frac{5 - \sqrt{91} + 12}{3} ((\frac{5 - \sqrt{91}}{3}) - 2)$

Step 8.1.2.7.2

Add $5$ and $12$ .

$\frac{- 16 - \sqrt{91}}{3} \cdot \frac{17 - \sqrt{91}}{3} ((\frac{5 - \sqrt{91}}{3}) - 2)$

Step 8.1.2.8

Multiply $\frac{- 16 - \sqrt{91}}{3} \cdot \frac{17 - \sqrt{91}}{3}$ .

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Step 8.1.2.8.1

Multiply $\frac{- 16 - \sqrt{91}}{3}$ by $\frac{17 - \sqrt{91}}{3}$ .

$\frac{(- 16 - \sqrt{91}) (17 - \sqrt{91})}{3 \cdot 3} ((\frac{5 - \sqrt{91}}{3}) - 2)$

Step 8.1.2.8.2

Multiply $3$ by $3$ .

$\frac{(- 16 - \sqrt{91}) (17 - \sqrt{91})}{9} ((\frac{5 - \sqrt{91}}{3}) - 2)$

Step 8.1.2.9

Expand $(- 16 - \sqrt{91}) (17 - \sqrt{91})$ using the FOIL Method.

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Step 8.1.2.9.1

Apply the distributive property.

$\frac{- 16 (17 - \sqrt{91}) - \sqrt{91} (17 - \sqrt{91})}{9} ((\frac{5 - \sqrt{91}}{3}) - 2)$

Step 8.1.2.9.2

Apply the distributive property.

$\frac{- 16 \cdot 17 - 16 (- \sqrt{91}) - \sqrt{91} (17 - \sqrt{91})}{9} ((\frac{5 - \sqrt{91}}{3}) - 2)$

Step 8.1.2.9.3

Apply the distributive property.

$\frac{- 16 \cdot 17 - 16 (- \sqrt{91}) - \sqrt{91} \cdot 17 - \sqrt{91} (- \sqrt{91})}{9} ((\frac{5 - \sqrt{91}}{3}) - 2)$

Step 8.1.2.10

Simplify and combine like terms.

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Step 8.1.2.10.1

Simplify each term.

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Step 8.1.2.10.1.1

Multiply $- 16$ by $17$ .

$\frac{- 272 - 16 (- \sqrt{91}) - \sqrt{91} \cdot 17 - \sqrt{91} (- \sqrt{91})}{9} ((\frac{5 - \sqrt{91}}{3}) - 2)$

Step 8.1.2.10.1.2

Multiply $- 1$ by $- 16$ .

$\frac{- 272 + 16 \sqrt{91} - \sqrt{91} \cdot 17 - \sqrt{91} (- \sqrt{91})}{9} ((\frac{5 - \sqrt{91}}{3}) - 2)$

Step 8.1.2.10.1.3

Multiply $17$ by $- 1$ .

$\frac{- 272 + 16 \sqrt{91} - 17 \sqrt{91} - \sqrt{91} (- \sqrt{91})}{9} ((\frac{5 - \sqrt{91}}{3}) - 2)$

Step 8.1.2.10.1.4

Multiply $- \sqrt{91} (- \sqrt{91})$ .

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Step 8.1.2.10.1.4.1

Multiply $- 1$ by $- 1$ .

$\frac{- 272 + 16 \sqrt{91} - 17 \sqrt{91} + 1 \sqrt{91} \sqrt{91}}{9} ((\frac{5 - \sqrt{91}}{3}) - 2)$

Step 8.1.2.10.1.4.2

Multiply $\sqrt{91}$ by $1$ .

$\frac{- 272 + 16 \sqrt{91} - 17 \sqrt{91} + \sqrt{91} \sqrt{91}}{9} ((\frac{5 - \sqrt{91}}{3}) - 2)$

Step 8.1.2.10.1.4.3

Raise $\sqrt{91}$ to the power of $1$ .

$\frac{- 272 + 16 \sqrt{91} - 17 \sqrt{91} + {\sqrt{91}}^{1} \sqrt{91}}{9} ((\frac{5 - \sqrt{91}}{3}) - 2)$

Step 8.1.2.10.1.4.4

Raise $\sqrt{91}$ to the power of $1$ .

$\frac{- 272 + 16 \sqrt{91} - 17 \sqrt{91} + {\sqrt{91}}^{1} {\sqrt{91}}^{1}}{9} ((\frac{5 - \sqrt{91}}{3}) - 2)$

Step 8.1.2.10.1.4.5

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

$\frac{- 272 + 16 \sqrt{91} - 17 \sqrt{91} + {\sqrt{91}}^{1 + 1}}{9} ((\frac{5 - \sqrt{91}}{3}) - 2)$

Step 8.1.2.10.1.4.6

Add $1$ and $1$ .

$\frac{- 272 + 16 \sqrt{91} - 17 \sqrt{91} + {\sqrt{91}}^{2}}{9} ((\frac{5 - \sqrt{91}}{3}) - 2)$

Step 8.1.2.10.1.5

Rewrite ${\sqrt{91}}^{2}$ as $91$ .

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Step 8.1.2.10.1.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{91}$ as $91^{\frac{1}{2}}$ .

$\frac{- 272 + 16 \sqrt{91} - 17 \sqrt{91} + {(91^{\frac{1}{2}})}^{2}}{9} ((\frac{5 - \sqrt{91}}{3}) - 2)$

Step 8.1.2.10.1.5.2

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

$\frac{- 272 + 16 \sqrt{91} - 17 \sqrt{91} + 91^{\frac{1}{2} \cdot 2}}{9} ((\frac{5 - \sqrt{91}}{3}) - 2)$

Step 8.1.2.10.1.5.3

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

$\frac{- 272 + 16 \sqrt{91} - 17 \sqrt{91} + 91^{\frac{2}{2}}}{9} ((\frac{5 - \sqrt{91}}{3}) - 2)$

Step 8.1.2.10.1.5.4

Cancel the common factor of $2$ .

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Step 8.1.2.10.1.5.4.1

Cancel the common factor.

$\frac{- 272 + 16 \sqrt{91} - 17 \sqrt{91} + 91^{\frac{2}{2}}}{9} ((\frac{5 - \sqrt{91}}{3}) - 2)$

Step 8.1.2.10.1.5.4.2

Rewrite the expression.

$\frac{- 272 + 16 \sqrt{91} - 17 \sqrt{91} + 91^{1}}{9} ((\frac{5 - \sqrt{91}}{3}) - 2)$

Step 8.1.2.10.1.5.5

Evaluate the exponent.

$\frac{- 272 + 16 \sqrt{91} - 17 \sqrt{91} + 91}{9} ((\frac{5 - \sqrt{91}}{3}) - 2)$

Step 8.1.2.10.2

Add $- 272$ and $91$ .

$\frac{- 181 + 16 \sqrt{91} - 17 \sqrt{91}}{9} ((\frac{5 - \sqrt{91}}{3}) - 2)$

Step 8.1.2.10.3

Subtract $17 \sqrt{91}$ from $16 \sqrt{91}$ .

$\frac{- 181 - \sqrt{91}}{9} ((\frac{5 - \sqrt{91}}{3}) - 2)$

Step 8.1.2.11

To write $- 2$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{- 181 - \sqrt{91}}{9} (\frac{5 - \sqrt{91}}{3} - 2 \cdot \frac{3}{3})$

Step 8.1.2.12

Combine fractions.

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Step 8.1.2.12.1

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

$\frac{- 181 - \sqrt{91}}{9} (\frac{5 - \sqrt{91}}{3} + \frac{- 2 \cdot 3}{3})$

Step 8.1.2.12.2

Combine the numerators over the common denominator.

$\frac{- 181 - \sqrt{91}}{9} \cdot \frac{5 - \sqrt{91} - 2 \cdot 3}{3}$

Step 8.1.2.13

Simplify the numerator.

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Step 8.1.2.13.1

Multiply $- 2$ by $3$ .

$\frac{- 181 - \sqrt{91}}{9} \cdot \frac{5 - \sqrt{91} - 6}{3}$

Step 8.1.2.13.2

Subtract $6$ from $5$ .

$\frac{- 181 - \sqrt{91}}{9} \cdot \frac{- 1 - \sqrt{91}}{3}$

Step 8.1.2.14

Multiply $\frac{- 181 - \sqrt{91}}{9} \cdot \frac{- 1 - \sqrt{91}}{3}$ .

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Step 8.1.2.14.1

Multiply $\frac{- 181 - \sqrt{91}}{9}$ by $\frac{- 1 - \sqrt{91}}{3}$ .

$\frac{(- 181 - \sqrt{91}) (- 1 - \sqrt{91})}{9 \cdot 3}$

Step 8.1.2.14.2

Multiply $9$ by $3$ .

$\frac{(- 181 - \sqrt{91}) (- 1 - \sqrt{91})}{27}$

Step 8.1.2.15

Expand $(- 181 - \sqrt{91}) (- 1 - \sqrt{91})$ using the FOIL Method.

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Step 8.1.2.15.1

Apply the distributive property.

$\frac{- 181 (- 1 - \sqrt{91}) - \sqrt{91} (- 1 - \sqrt{91})}{27}$

Step 8.1.2.15.2

Apply the distributive property.

$\frac{- 181 \cdot - 1 - 181 (- \sqrt{91}) - \sqrt{91} (- 1 - \sqrt{91})}{27}$

Step 8.1.2.15.3

Apply the distributive property.

$\frac{- 181 \cdot - 1 - 181 (- \sqrt{91}) - \sqrt{91} \cdot - 1 - \sqrt{91} (- \sqrt{91})}{27}$

Step 8.1.2.16

Simplify and combine like terms.

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Step 8.1.2.16.1

Simplify each term.

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Step 8.1.2.16.1.1

Multiply $- 181$ by $- 1$ .

$\frac{181 - 181 (- \sqrt{91}) - \sqrt{91} \cdot - 1 - \sqrt{91} (- \sqrt{91})}{27}$

Step 8.1.2.16.1.2

Multiply $- 1$ by $- 181$ .

$\frac{181 + 181 \sqrt{91} - \sqrt{91} \cdot - 1 - \sqrt{91} (- \sqrt{91})}{27}$

Step 8.1.2.16.1.3

Multiply $- \sqrt{91} \cdot - 1$ .

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Step 8.1.2.16.1.3.1

Multiply $- 1$ by $- 1$ .

$\frac{181 + 181 \sqrt{91} + 1 \sqrt{91} - \sqrt{91} (- \sqrt{91})}{27}$

Step 8.1.2.16.1.3.2

Multiply $\sqrt{91}$ by $1$ .

$\frac{181 + 181 \sqrt{91} + \sqrt{91} - \sqrt{91} (- \sqrt{91})}{27}$

Step 8.1.2.16.1.4

Multiply $- \sqrt{91} (- \sqrt{91})$ .

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Step 8.1.2.16.1.4.1

Multiply $- 1$ by $- 1$ .

$\frac{181 + 181 \sqrt{91} + \sqrt{91} + 1 \sqrt{91} \sqrt{91}}{27}$

Step 8.1.2.16.1.4.2

Multiply $\sqrt{91}$ by $1$ .

$\frac{181 + 181 \sqrt{91} + \sqrt{91} + \sqrt{91} \sqrt{91}}{27}$

Step 8.1.2.16.1.4.3

Raise $\sqrt{91}$ to the power of $1$ .

$\frac{181 + 181 \sqrt{91} + \sqrt{91} + {\sqrt{91}}^{1} \sqrt{91}}{27}$

Step 8.1.2.16.1.4.4

Raise $\sqrt{91}$ to the power of $1$ .

$\frac{181 + 181 \sqrt{91} + \sqrt{91} + {\sqrt{91}}^{1} {\sqrt{91}}^{1}}{27}$

Step 8.1.2.16.1.4.5

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

$\frac{181 + 181 \sqrt{91} + \sqrt{91} + {\sqrt{91}}^{1 + 1}}{27}$

Step 8.1.2.16.1.4.6

Add $1$ and $1$ .

$\frac{181 + 181 \sqrt{91} + \sqrt{91} + {\sqrt{91}}^{2}}{27}$

Step 8.1.2.16.1.5

Rewrite ${\sqrt{91}}^{2}$ as $91$ .

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Step 8.1.2.16.1.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{91}$ as $91^{\frac{1}{2}}$ .

$\frac{181 + 181 \sqrt{91} + \sqrt{91} + {(91^{\frac{1}{2}})}^{2}}{27}$

Step 8.1.2.16.1.5.2

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

$\frac{181 + 181 \sqrt{91} + \sqrt{91} + 91^{\frac{1}{2} \cdot 2}}{27}$

Step 8.1.2.16.1.5.3

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

$\frac{181 + 181 \sqrt{91} + \sqrt{91} + 91^{\frac{2}{2}}}{27}$

Step 8.1.2.16.1.5.4

Cancel the common factor of $2$ .

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Step 8.1.2.16.1.5.4.1

Cancel the common factor.

$\frac{181 + 181 \sqrt{91} + \sqrt{91} + 91^{\frac{2}{2}}}{27}$

Step 8.1.2.16.1.5.4.2

Rewrite the expression.

$\frac{181 + 181 \sqrt{91} + \sqrt{91} + 91^{1}}{27}$

Step 8.1.2.16.1.5.5

Evaluate the exponent.

$\frac{181 + 181 \sqrt{91} + \sqrt{91} + 91}{27}$

Step 8.1.2.16.2

Add $181$ and $91$ .

$\frac{272 + 181 \sqrt{91} + \sqrt{91}}{27}$

Step 8.1.2.16.3

Add $181 \sqrt{91}$ and $\sqrt{91}$ .

$\frac{272 + 182 \sqrt{91}}{27}$

Step 8.2

Write the $x$ and $y$ coordinates in point form.

$(\frac{5 - \sqrt{91}}{3}, \frac{272 + 182 \sqrt{91}}{27})$

Step 9

Since the first derivative changed signs from negative to positive around $x = \frac{5 + \sqrt{91}}{3}$ , then there is a turning point at $x = \frac{5 + \sqrt{91}}{3}$ .

Step 10

Find the y-coordinate of $\frac{5 + \sqrt{91}}{3}$ to find the turning point.

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Step 10.1

Find $p (\frac{5 + \sqrt{91}}{3})$ to find the y-coordinate of $\frac{5 + \sqrt{91}}{3}$ .

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Step 10.1.1

Replace the variable $x$ with $\frac{5 + \sqrt{91}}{3}$ in the expression.

$p (\frac{5 + \sqrt{91}}{3}) = ((\frac{5 + \sqrt{91}}{3}) - 7) ((\frac{5 + \sqrt{91}}{3}) + 4) ((\frac{5 + \sqrt{91}}{3}) - 2)$

Step 10.1.2

Simplify $((\frac{5 + \sqrt{91}}{3}) - 7) ((\frac{5 + \sqrt{91}}{3}) + 4) ((\frac{5 + \sqrt{91}}{3}) - 2)$ .

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Step 10.1.2.1

Remove parentheses.

$((\frac{5 + \sqrt{91}}{3}) - 7) ((\frac{5 + \sqrt{91}}{3}) + 4) ((\frac{5 + \sqrt{91}}{3}) - 2)$

Step 10.1.2.2

To write $- 7$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$(\frac{5 + \sqrt{91}}{3} - 7 \cdot \frac{3}{3}) ((\frac{5 + \sqrt{91}}{3}) + 4) ((\frac{5 + \sqrt{91}}{3}) - 2)$

Step 10.1.2.3

Combine fractions.

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Step 10.1.2.3.1

Combine $- 7$ and $\frac{3}{3}$ .

$(\frac{5 + \sqrt{91}}{3} + \frac{- 7 \cdot 3}{3}) ((\frac{5 + \sqrt{91}}{3}) + 4) ((\frac{5 + \sqrt{91}}{3}) - 2)$

Step 10.1.2.3.2

Combine the numerators over the common denominator.

$\frac{5 + \sqrt{91} - 7 \cdot 3}{3} ((\frac{5 + \sqrt{91}}{3}) + 4) ((\frac{5 + \sqrt{91}}{3}) - 2)$

Step 10.1.2.4

Simplify the numerator.

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Step 10.1.2.4.1

Multiply $- 7$ by $3$ .

$\frac{5 + \sqrt{91} - 21}{3} ((\frac{5 + \sqrt{91}}{3}) + 4) ((\frac{5 + \sqrt{91}}{3}) - 2)$

Step 10.1.2.4.2

Subtract $21$ from $5$ .

$\frac{- 16 + \sqrt{91}}{3} ((\frac{5 + \sqrt{91}}{3}) + 4) ((\frac{5 + \sqrt{91}}{3}) - 2)$

Step 10.1.2.5

To write $4$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{- 16 + \sqrt{91}}{3} (\frac{5 + \sqrt{91}}{3} + 4 \cdot \frac{3}{3}) ((\frac{5 + \sqrt{91}}{3}) - 2)$

Step 10.1.2.6

Combine fractions.

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Step 10.1.2.6.1

Combine $4$ and $\frac{3}{3}$ .

$\frac{- 16 + \sqrt{91}}{3} (\frac{5 + \sqrt{91}}{3} + \frac{4 \cdot 3}{3}) ((\frac{5 + \sqrt{91}}{3}) - 2)$

Step 10.1.2.6.2

Combine the numerators over the common denominator.

$\frac{- 16 + \sqrt{91}}{3} \cdot \frac{5 + \sqrt{91} + 4 \cdot 3}{3} ((\frac{5 + \sqrt{91}}{3}) - 2)$

Step 10.1.2.7

Simplify the numerator.

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Step 10.1.2.7.1

Multiply $4$ by $3$ .

$\frac{- 16 + \sqrt{91}}{3} \cdot \frac{5 + \sqrt{91} + 12}{3} ((\frac{5 + \sqrt{91}}{3}) - 2)$

Step 10.1.2.7.2

Add $5$ and $12$ .

$\frac{- 16 + \sqrt{91}}{3} \cdot \frac{17 + \sqrt{91}}{3} ((\frac{5 + \sqrt{91}}{3}) - 2)$

Step 10.1.2.8

Multiply $\frac{- 16 + \sqrt{91}}{3} \cdot \frac{17 + \sqrt{91}}{3}$ .

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Step 10.1.2.8.1

Multiply $\frac{- 16 + \sqrt{91}}{3}$ by $\frac{17 + \sqrt{91}}{3}$ .

$\frac{(- 16 + \sqrt{91}) (17 + \sqrt{91})}{3 \cdot 3} ((\frac{5 + \sqrt{91}}{3}) - 2)$

Step 10.1.2.8.2

Multiply $3$ by $3$ .

$\frac{(- 16 + \sqrt{91}) (17 + \sqrt{91})}{9} ((\frac{5 + \sqrt{91}}{3}) - 2)$

Step 10.1.2.9

Expand $(- 16 + \sqrt{91}) (17 + \sqrt{91})$ using the FOIL Method.

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Step 10.1.2.9.1

Apply the distributive property.

$\frac{- 16 (17 + \sqrt{91}) + \sqrt{91} (17 + \sqrt{91})}{9} ((\frac{5 + \sqrt{91}}{3}) - 2)$

Step 10.1.2.9.2

Apply the distributive property.

$\frac{- 16 \cdot 17 - 16 \sqrt{91} + \sqrt{91} (17 + \sqrt{91})}{9} ((\frac{5 + \sqrt{91}}{3}) - 2)$

Step 10.1.2.9.3

Apply the distributive property.

$\frac{- 16 \cdot 17 - 16 \sqrt{91} + \sqrt{91} \cdot 17 + \sqrt{91} \sqrt{91}}{9} ((\frac{5 + \sqrt{91}}{3}) - 2)$

Step 10.1.2.10

Simplify and combine like terms.

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Step 10.1.2.10.1

Simplify each term.

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Step 10.1.2.10.1.1

Multiply $- 16$ by $17$ .

$\frac{- 272 - 16 \sqrt{91} + \sqrt{91} \cdot 17 + \sqrt{91} \sqrt{91}}{9} ((\frac{5 + \sqrt{91}}{3}) - 2)$

Step 10.1.2.10.1.2

Move $17$ to the left of $\sqrt{91}$ .

$\frac{- 272 - 16 \sqrt{91} + 17 \cdot \sqrt{91} + \sqrt{91} \sqrt{91}}{9} ((\frac{5 + \sqrt{91}}{3}) - 2)$

Step 10.1.2.10.1.3

Combine using the product rule for radicals.

$\frac{- 272 - 16 \sqrt{91} + 17 \sqrt{91} + \sqrt{91 \cdot 91}}{9} ((\frac{5 + \sqrt{91}}{3}) - 2)$

Step 10.1.2.10.1.4

Multiply $91$ by $91$ .

$\frac{- 272 - 16 \sqrt{91} + 17 \sqrt{91} + \sqrt{8281}}{9} ((\frac{5 + \sqrt{91}}{3}) - 2)$

Step 10.1.2.10.1.5

Rewrite $8281$ as $91^{2}$ .

$\frac{- 272 - 16 \sqrt{91} + 17 \sqrt{91} + \sqrt{91^{2}}}{9} ((\frac{5 + \sqrt{91}}{3}) - 2)$

Step 10.1.2.10.1.6

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

$\frac{- 272 - 16 \sqrt{91} + 17 \sqrt{91} + 91}{9} ((\frac{5 + \sqrt{91}}{3}) - 2)$

Step 10.1.2.10.2

Add $- 272$ and $91$ .

$\frac{- 181 - 16 \sqrt{91} + 17 \sqrt{91}}{9} ((\frac{5 + \sqrt{91}}{3}) - 2)$

Step 10.1.2.10.3

Add $- 16 \sqrt{91}$ and $17 \sqrt{91}$ .

$\frac{- 181 + \sqrt{91}}{9} ((\frac{5 + \sqrt{91}}{3}) - 2)$

Step 10.1.2.11

To write $- 2$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{- 181 + \sqrt{91}}{9} (\frac{5 + \sqrt{91}}{3} - 2 \cdot \frac{3}{3})$

Step 10.1.2.12

Combine fractions.

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Step 10.1.2.12.1

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

$\frac{- 181 + \sqrt{91}}{9} (\frac{5 + \sqrt{91}}{3} + \frac{- 2 \cdot 3}{3})$

Step 10.1.2.12.2

Combine the numerators over the common denominator.

$\frac{- 181 + \sqrt{91}}{9} \cdot \frac{5 + \sqrt{91} - 2 \cdot 3}{3}$

Step 10.1.2.13

Simplify the numerator.

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Step 10.1.2.13.1

Multiply $- 2$ by $3$ .

$\frac{- 181 + \sqrt{91}}{9} \cdot \frac{5 + \sqrt{91} - 6}{3}$

Step 10.1.2.13.2

Subtract $6$ from $5$ .

$\frac{- 181 + \sqrt{91}}{9} \cdot \frac{- 1 + \sqrt{91}}{3}$

Step 10.1.2.14

Multiply $\frac{- 181 + \sqrt{91}}{9} \cdot \frac{- 1 + \sqrt{91}}{3}$ .

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Step 10.1.2.14.1

Multiply $\frac{- 181 + \sqrt{91}}{9}$ by $\frac{- 1 + \sqrt{91}}{3}$ .

$\frac{(- 181 + \sqrt{91}) (- 1 + \sqrt{91})}{9 \cdot 3}$

Step 10.1.2.14.2

Multiply $9$ by $3$ .

$\frac{(- 181 + \sqrt{91}) (- 1 + \sqrt{91})}{27}$

Step 10.1.2.15

Expand $(- 181 + \sqrt{91}) (- 1 + \sqrt{91})$ using the FOIL Method.

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Step 10.1.2.15.1

Apply the distributive property.

$\frac{- 181 (- 1 + \sqrt{91}) + \sqrt{91} (- 1 + \sqrt{91})}{27}$

Step 10.1.2.15.2

Apply the distributive property.

$\frac{- 181 \cdot - 1 - 181 \sqrt{91} + \sqrt{91} (- 1 + \sqrt{91})}{27}$

Step 10.1.2.15.3

Apply the distributive property.

$\frac{- 181 \cdot - 1 - 181 \sqrt{91} + \sqrt{91} \cdot - 1 + \sqrt{91} \sqrt{91}}{27}$

Step 10.1.2.16

Simplify and combine like terms.

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Step 10.1.2.16.1

Simplify each term.

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Step 10.1.2.16.1.1

Multiply $- 181$ by $- 1$ .

$\frac{181 - 181 \sqrt{91} + \sqrt{91} \cdot - 1 + \sqrt{91} \sqrt{91}}{27}$

Step 10.1.2.16.1.2

Move $- 1$ to the left of $\sqrt{91}$ .

$\frac{181 - 181 \sqrt{91} - 1 \cdot \sqrt{91} + \sqrt{91} \sqrt{91}}{27}$

Step 10.1.2.16.1.3

Rewrite $- 1 \sqrt{91}$ as $- \sqrt{91}$ .

$\frac{181 - 181 \sqrt{91} - \sqrt{91} + \sqrt{91} \sqrt{91}}{27}$

Step 10.1.2.16.1.4

Combine using the product rule for radicals.

$\frac{181 - 181 \sqrt{91} - \sqrt{91} + \sqrt{91 \cdot 91}}{27}$

Step 10.1.2.16.1.5

Multiply $91$ by $91$ .

$\frac{181 - 181 \sqrt{91} - \sqrt{91} + \sqrt{8281}}{27}$

Step 10.1.2.16.1.6

Rewrite $8281$ as $91^{2}$ .

$\frac{181 - 181 \sqrt{91} - \sqrt{91} + \sqrt{91^{2}}}{27}$

Step 10.1.2.16.1.7

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

$\frac{181 - 181 \sqrt{91} - \sqrt{91} + 91}{27}$

Step 10.1.2.16.2

Add $181$ and $91$ .

$\frac{272 - 181 \sqrt{91} - \sqrt{91}}{27}$

Step 10.1.2.16.3

Subtract $\sqrt{91}$ from $- 181 \sqrt{91}$ .

$\frac{272 - 182 \sqrt{91}}{27}$

Step 10.2

Write the $x$ and $y$ coordinates in point form.

$(\frac{5 + \sqrt{91}}{3}, \frac{272 - 182 \sqrt{91}}{27})$

Step 11

These are the turning points.

$(\frac{5 - \sqrt{91}}{3}, \frac{272 + 182 \sqrt{91}}{27})$

$(\frac{5 + \sqrt{91}}{3}, \frac{272 - 182 \sqrt{91}}{27})$

Step 12