Answer the following questions for the function f(x) = x^3/x^2 - 1 defined on th

Question

Answer the following questions for the function f(x) = x^3/x^2 - 1 defined on the interval [-20, 20]. Enter the x-coordinates of the vertical asymptotes of f(x) as a comma-separated list. That is, if there is just one value, give it: if there are more than one, enter them separated commas: and if there are none, enter NONE. Answer f(x) Is concave up on the region Note: Give your answer in interval notation. Enter the x-coordinates of the inflection point(s) for this function as a comma-separated list. Answer:

Explanation / Answer

(a) x^2 - 1 = 0 means x^2 = 1 so x = {-1, 1}

(b) f'(x) = [(x^2 - 1)(3x^2) - (x^3)(2x)] / (x^2 - 1)^2 = (x^2)(x^2 - 3) / (x^2 - 1)^2

f"(x) = [2x(x^2 + 3)] / (x^2 - 1)^3

f"(x) > 0 implies [2x(x^2 + 3)] / (x^2 - 1)^3 > 0, which gives -1 < x < 0 or x > 1, that is (-1, 0) U (1, )

(c) [2x(x^2 + 3)] / (x^2 - 1)^3 = 0 for x = 0

The function has an inflection point at x = 0.

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