Graphs of Increasing and Decreasing Functions and Asymptotes Practice Questions

f(x) has no asymptotes.

f'(x) = 2x – 30, thus there is a local minimum at (15,–210). Because f"(x) = 2, the graph is always concave up.

g(x) has no asymptotes.

g'(x) = –4 – 2x = –2(2 + x), so there is a local maximum at (–2,4). Because g"(x) = –2, the graph is always concave down.

h(x) has no asymptotes.

h'(x) = 6x² – 6x – 36 = 6(x – 3)(x + 2), so there is a local maximum at (–2,49) and a local minimum at (3,–76). Because h"(x) = 12x – 6 = 12 (x – ) there is a point of inflection at (, – 13).

k(x) has no asymptotes.

k'(x) = 3 – 3x² = 3(1 – x)(1 + x), so there is a local minimum at (–1,–2) and a local maximum at (1,2). k"(x) = –6x, so there is a point of inflection at (0,0).

f(x) has no asymptotes.

f'(x) = 4x³ = 24x² = 4x²(x – 6), so there is a local minimum at (6, –427).

f"(x) = 12x² – 48x = 12x(x – 4), so there are points of inflection at (0, – 5) and at (4, –251).

g(x) has a vertical asymptote at x = –2 and a horizontal asymptote at y = 1. The first derivative is g'(x) = , and the second is g''(x) =

h(x) = has vertical asymptotes at x = – 3 and x = 3, and a horizontal asymptote at y = 0. Because h '(x) = , there is a local maximum at . The second derivative is h '' (x) = .

has vertical asymptotes at x = 1 and x = -1, and a horizontal asymptote at y = 0. The first derivative is , and the second derivative is . There is a point of inflection at (0,0).

j(x) has a vertical asymptote at x = 0 but no horizontal asymptotes. Because j'(x) = = , there is a local maximum at (–1,–2) and a local minimum at (1,2). The second derivative is j"( x) =

f(x) has a horizontal asymptote at y = 0 but no vertical asymptotes. Because f(x) = = , there is a local minimum at (–1,–) and a local maximum at (1,) Because f'' (x) = = , there are points of inflection at ( ), (0,0), and ().