- 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.