Question

6. Sketch a single graph with the following characteristics. a. dom(1) = (0,0). b. f has zeros at x = -5 and x = 8. c. f has a horizontal asymptote at y = 2. d. f'(x) > 0 on (-5,2) and (8,00). e. f'(x) < 0 on (-00, -5) and (2,8). f. f" (x) > 0 on (-8, 2) and (2,00). g. f"(x) < 0 on (-0,–8).

Answer 1

To sketch the graph, the following concepts from calculus are essential :

f has an asymptote at y = 2 implies that the line y = 2 is tangent to the graph of f as either x tends to $\infty$ or as x tends to
OX

f'(x) > 0 implies that f is strictly increasing on that domain.
f'(x) < 0 implies that f is strictly decreasing on that domain.
f''(x) > 0 implies that f is concave upward on that domain.
f''(x) < 0 implies that f is concave downward on that domain.

- Assymptote y az 2 + -8 - 5 2 8