Math 112 Solutions for 3.2, p. 237

1.
(a) none. (c) 0, . (d) . (e) o, 3, -3. (g) 1. (i) . (k) . (o) 0.
2.
(a) increasing and concave upward. (b) concave downward. (c) decreasing and concave upward. (d) increasing and concave downward. (e) decreasing and concave downward.
6.
Since is of odd degree, it has at least one real zero. If has two real zeros, then has a critical point, by Rolle's Theorem. But for every , and has no critical point, so at most one real zero.
8.
(c)
9.
Let be the positions of the cars, and we assume and are differentiable. Let . Since each car passes the other, there are two times and at which they have the same position, so . Hence by Rolle's Theorem there is a time between and such that , so that and the cars have the same speed at time .
10.
Since the cars pass each other, there is a time at which their speeds are the same; when they pass each other again, there is a time at which their speeds are the same. Hence there is a time between and when the derivatives of their speeds, their accelerations, are the same.
11.
(a) (c) (d) .
12.
(d) is not differentiable at 0.
13.
(c) cp: 0, 2. hcp: 1. increasing in and in . decreasing in . concave down in . concave up in . infl.pt. at 1

(g) cp: -1, 1. hcp: none (0 is not in the domain). increasing in and in . decreasing in (-1, 0) and in (0, 1). concave down in . concave up in . infl.pt.: none.

(h) cp: 0. increasing in , decreasing in . hcp: . concave down in , concave up in and in .

(l) cp: none. increasing everywhere. hcp: . concave down in , concave up in .

(n) cp: 1. increasing in , decreasing in . hcp: 2. concave down in , concave up in .

(o) cp: 0. hcp: -1, 1. increasing in . decreasing in . concave up in and in . concave down in (-1, 1). infl.pt.: -1, 1.

14.
(c) vertical tangent line at 0.
15.
If , then is decreasing by Theorem 71, so is concave downward.
16.
(a) . (b) . (c) The curve is concave upward, so the tangent line, and hence its intercept, is always below the curve.
17.
(a) If , then by Rolle's Theorem there is a point between and at which . But then is a critical point, a contradiction.
19.
(c) cusp. (d) vertical tangent line.
21.
(a) The curve is concave downward because as the temperature increases, the rate of temperature increase lessens. (b) . (c) . (d) between and .

22.
Apply the Mean Value Theorem to on the interval . Thus there exists such that , from which the result follows.