Answer Key
821
x < − 1
x = − 1
231 . a. Increases over − 1 4 <
x < 3
x > 3
x = 3 4
4 , decreases over
4 and
4 b. Minimum at
4 , maximumat
c. Concave up for − 3 4 < x = 1 4 233 . a. Increasing for all x b. No local minimum or maximum c. Concave up for x >0, concave down for x <0 d. Inflection point at x =0 235 . a. Increasing for all x where defined b. No local minima or maxima c. Concave up for x <1; concave down for x >1 d. No inflection points in domain 237 . a. Increasing over − π 4 < x < 3 π 4 , decreasing over x > 3 π 4 , x < − π 4 b. Minimum at x = − π 4 , maximumat x = 3 π 4 c. Concave up for − π 2 < x < π 2 , concave down for x < − π 2 , x > π 2 d. Infection points at x =± π 2 239 . a. Increasing over x >4, decreasing over 0< x <4 b. Minimum at x =4 c. Concave up for 0< x <8 2 3 , concave down for x >8 2 3 d. Inflection point at x =8 2 3 x < 1 4 , concave down for x < − 3 4 and x > 1 4 d. Inflection points at x = − 3 4 ,
241 . f >0, f ′ >0, f ″<0 243 . f >0, f ′ <0, f ″<0 245 . f >0, f ′ >0, f ″>0 247 . True, by the Mean Value Theorem 249 . True, examine derivative 251 . x =1 253 . x =−1, x =2 255 . x =0 257 . Yes, there is a vertical asymptote 259 . Yes, there is vertical asymptote 261 . 0 263 . ∞
265 . − 1 7 267 . −2 269 . −4
271 . Horizontal: none, vertical: x =0 273 . Horizontal: none, vertical: x =±2 275 . Horizontal: none, vertical: none 277 . Horizontal: y =0, vertical: x =±1 279 . Horizontal: y =0, vertical: x =0 and x =−1 281 . Horizontal: y =1, vertical: x =1 283 . Horizontal: none, vertical: none 285 . Answers will vary, for example: y = 2 x x −1 287 . Answers will vary, for example: y = 4 x x +1 289 . y =0 291 . ∞ 293 . y =3 295 .
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