Suppose the function g(x) has a domain of all real numbers except x = -5. The second derivative...

Question:

Suppose the function g(x) has a domain of all real numbers except x = -5. The second derivative of g(x) is {eq}\displaystyle g''(x) = \frac{(x-4) (x+3)}{(x-5)^{5}} {/eq}

A) Give the intervals where g(x) is concave up.

B) Give the intervals where g(x) is concave down.

C) Find the x-coordinates of the inflection points for g(x).

Concavity of Function:

We have to understand the concavity of a function for solving this problem:

Concave up: {eq}y = f(x) {/eq} is concave up on an interval {eq}I {/eq} when {eq}\displaystyle f''(x) > 0 {/eq} in the interval {eq}I {/eq}.

Concave down: {eq}y = f(x) {/eq} is concave down on an interval {eq}I {/eq} when {eq}\displaystyle f''(x) < 0 {/eq} in the interval {eq}I {/eq}.

Answer and Explanation:

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Given:

{eq}\displaystyle g''(x) = \frac{(x-4) (x+3)}{(x-5)^{5}} {/eq}

First we will draw sign scheme of {eq}g''(x) {/eq} on the number line.

So...

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Concavity and Inflection Points on Graphs

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Chapter 9 / Lesson 5
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