# What are the critical values, if any, of f(x)= (x^2+6x-7)^2?

Nov 19, 2015

The critical values for $f$ are $- 7 , - 3 , \text{and } 1$

#### Explanation:

Use the power rule and the chain rule to find:

$f ' \left(x\right) = 2 \left({x}^{2} + 6 x - 7\right) \left(2 x + 6\right)$.

Since, $f \left(x\right)$ and $f ' \left(x\right)$ are polynomials, the critical values for $f$ are the zeros of $f '$

Solve $f ' \left(x\right) = 2 \left({x}^{2} + 6 x - 7\right) \left(2 x + 6\right) = 0$ by solving

${x}^{2} + 6 x - 7 = 0$ or $2 x + 6 = 0$. So,

$\left(x + 7\right) \left(x - 1\right) = 0$ or $x = - 3$

$x = - 7 , 1 , \text{or} - 3$

The critical values for $f$ are $- 7 , - 3 , \text{and } 1$
The text passage contains a mathematical question about finding the critical values of a given function, and it provides a detailed solution to the problem.
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Given a text passage, determine:
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First, describe the content structure you found in 1-2 sentences.

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TYPE: <one of the following values>
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# What are the critical values, if any, of f(x)= (x^2+6x-7)^2?

Nov 19, 2015

The critical values for $f$ are $- 7 , - 3 , \text{and } 1$

#### Explanation:

Use the power rule and the chain rule to find:

$f ' \left(x\right) = 2 \left({x}^{2} + 6 x - 7\right) \left(2 x + 6\right)$.

Since, $f \left(x\right)$ and $f ' \left(x\right)$ are polynomials, the critical values for $f$ are the zeros of $f '$

Solve $f ' \left(x\right) = 2 \left({x}^{2} + 6 x - 7\right) \left(2 x + 6\right) = 0$ by solving

${x}^{2} + 6 x - 7 = 0$ or $2 x + 6 = 0$. So,

$\left(x + 7\right) \left(x - 1\right) = 0$ or $x = - 3$

$x = - 7 , 1 , \text{or} - 3$

The critical values for $f$ are $- 7 , - 3 , \text{and } 1$
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The text passage contains a mathematical question about finding the critical values of a given function, and it provides a detailed solution to the problem.

TYPE: SINGLE_QUESTION_WITH_ANSWER