Question

9. Consider the function f(x) given below. The first and second derivatives of f are also provided (you do not need to check them): f(x) = x/(x^3 - 1), f'(x) = -(2x^3 + 1)/(x^3 - 1)^2, f''(x) = (6x^2(x^3 + 2))/(x^3 - 1)^3 (a) Find the domain of f(x). (b) Find all asymptotes of f(x). Each vertical asymptote you find should be described by two one-sided limits, and each horizontal asymptote by one limit. (c) Find the intervals where f(x) is increasing or decreasing. (d) Find the x and y coordinates of all critical points of f(x) and indicate whether each is a local maximum, local minimum, or neither. If there are no critical points, say so. (e) Find the intervals where f is concave-up or concave-down. (f) Find the x and y coordinates of all points of inflection of f(x). If there are no points of inflection, say so. (g) Sketch the graph of f(x). Your graph should clearly reflect all the information you found in the previous parts.

          9. Consider the function f(x) given below. The first and second derivatives of f are also provided (you do not need to check them):

f(x) = x/(x^3 - 1), f'(x) = -(2x^3 + 1)/(x^3 - 1)^2, f''(x) = (6x^2(x^3 + 2))/(x^3 - 1)^3

(a) Find the domain of f(x).
(b) Find all asymptotes of f(x). Each vertical asymptote you find should be described by two one-sided limits, and each horizontal asymptote by one limit.
(c) Find the intervals where f(x) is increasing or decreasing.
(d) Find the x and y coordinates of all critical points of f(x) and indicate whether each is a local maximum, local minimum, or neither. If there are no critical points, say so.
(e) Find the intervals where f is concave-up or concave-down.
(f) Find the x and y coordinates of all points of inflection of f(x). If there are no points of inflection, say so.
(g) Sketch the graph of f(x). Your graph should clearly reflect all the information you found in the previous parts.

9. Consider the function f(x) given below. The first and second derivatives of f are also provided (you do not need to check them):

f(x) = x/(x^3 - 1), f'(x) = -(2x^3 + 1)/(x^3 - 1)^2, f”(x) = (6x^2(x^3 + 2))/(x^3 - 1)^3

(a) Find the domain of f(x).
(b) Find all asymptotes of f(x). Each vertical asymptote you find should be described by two one-sided limits, and each horizontal asymptote by one limit.
(c) Find the intervals where f(x) is increasing or decreasing.
(d) Find the x and y coordinates of all critical points of f(x) and indicate whether each is a local maximum, local minimum, or neither. If there are no critical points, say so.
(e) Find the intervals where f is concave-up or concave-down.
(f) Find the x and y coordinates of all points of inflection of f(x). If there are no points of inflection, say so.
(g) Sketch the graph of f(x). Your graph should clearly reflect all the information you found in the previous parts.

Added by Nerea S.

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