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The function f(x)= \frac {x^3}{x^2-16} defined on the interval [-19,16]. Enter points, such as...

Question:

The function {eq}\displaystyle f(x)= \frac {x^3}{x^2-16} {/eq} defined on the interval {eq}[-19,16] {/eq}. Enter points, such as inflection points in ascending order, i.e. smallest {eq}x {/eq} values first. Enter intervals in ascending order also.

a) What are {eq}f(x) {/eq} two vertical asymptotes?

b) What is the inflection point?

c) What is the concave up on the region for {eq}f(x) {/eq}?

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A Function with one independent variable has inflection point where its second derivative is zero, remember, if the second derivative is positive the function is concave up and if the second derivative is negative, the function is concave down. Also, vertical asymptote exists where the denominator is zero.

Answer and Explanation:

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The function is:

{eq}\displaystyle \ f(x) = \frac {x^3}{x^2-16} \\ \displaystyle \ f(x) = {\frac {{x}^{3}}{ \left( x-4 \right) \left( x+4...

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Concavity and Inflection Points on Graphs

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Chapter 9 / Lesson 5
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