5128_ch04ansTE_pp652-661
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Additional Answers 655<br />
35. y3x 2 6x and y6x 6.<br />
y0 at 2 and 0. y(2) 0 and y(0) 0, so there is a local maximum<br />
at (– 2, 2) and a local minimum at (0, 2).<br />
36. y15x 4 75x 2 60 and y60x 3 150x.<br />
y0 at 1 and 2. y(2) 0, y(1) 0, y(1) 0, and y(2) 0;<br />
so there are local maxima at (2, 4) and (1, 58), and there are local<br />
minima at (1, 18) and (2, 36).<br />
37. y(x 1)e x and y(x 2)e x .<br />
y0 at 1 and y(1) 0 , so there is a local minimum at (1, 1/e).<br />
38. y(1 x)e x and y(x 2)e x .<br />
y0 at 1 and y(1) 0, so there is a local maximum at (1, 1e)<br />
41.<br />
y<br />
y = f′(x)<br />
y = f(x)<br />
P<br />
47. One possible answer:<br />
y<br />
(–2, 8) 10<br />
(0, 4)<br />
–5<br />
(2, 0)<br />
–10<br />
48. One possible answer:<br />
5<br />
x<br />
x<br />
y = f′′(x)<br />
42.<br />
43. No. f must have a horizontal tangent line at that point, but it could be increasing<br />
(or decreasing) on both sides of the point, and there would be no<br />
local extremum.<br />
44. No. f (x) could still be positive (or negative) on both sides of x c, in<br />
which case the concavity of the function wouldn’t change at x c.<br />
45. One possible answer:<br />
y<br />
–5<br />
5<br />
5<br />
x<br />
51. (a) Absolute maximum at (1, 2);<br />
absolute minimum at (3, 2)<br />
(b) None<br />
(c) One possible answer:<br />
2<br />
1<br />
–1<br />
–2<br />
y<br />
1<br />
y = f(x)<br />
52. (a) Absolute maximum at (0, 2);<br />
absolute minimum at (2, 1) and (2, 1)<br />
(b) At (1, 0) and (1, 0)<br />
(c) One possible answer:<br />
2<br />
3<br />
x<br />
–5<br />
46. One possible answer:<br />
(d) f(3) f(3), and 1 f(3) 0.