f(x)=2x+8x^-1.For this function there are four important intervals: (-infinity,A],[A,B),(B,C),...

Question:

f(x)=2x+8x{eq}^{-1} {/eq}.For this function there are four important intervals: {eq}(-\infty,A],[A,B),(B,C), and [C,\infty) {/eq} where A,B and C are either critical numbers or points at which f(x) is undefined.

Find A,B and C.

Critical Numbers:

We will find the critical numbers of the function by differentiating the function and then putting it equal to 0 and we will also check the value where the function is not defined.

Answer and Explanation:

To find the critical numbers we will differentiate the equation:

{eq}f(x)=2x+\frac{8}{x} {/eq}

Differentiating it we get:

{eq}f'(x)=2-\frac{8}{x^{2}}=0\\ x=\pm 2 {/eq}

The function is not defined at x=0:

{eq}=(-\infty, -2), (-2,0), (0,2), (2, \infty) {/eq}


Learn more about this topic:

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Finding Critical Points in Calculus: Function & Graph

from CAHSEE Math Exam: Tutoring Solution

Chapter 8 / Lesson 9
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