Question

Find all critical numbers of the function f(x) = (x - 9). Then use the second-derivative test on each critical number to determine whether it leads to a local maximum or minimum Select the correct choice below and, if necessary, fill in the answer box(es) to complete your choice. O A. The critical number(s) is/are at x = There is no local maximum and no local minimum. (Type an integer or a simplified fraction. Use a comma to separate answers as needed.) OB. The critical number(s) is/are at x = There is a local maximum at x = There is a local minimum at x = (Type integers or simplified fractions. Use comma to separate answers as needed.) OC. The critical number(s) is/are at x = There is a local maximum at x = There is no local minimum. (Type integers or simplified fractions. Use a comma to separate answers as needed.) OD. The critical number(s) is/are at x = . There is no local maximum. There is a local minimum at x = (Type integers or simplified fractions. Use a comma to separate answers as needed.) O E. There are no critical numbers.

Answer 1

= 0. Now 5(2-1) 5 is oded for which fe (9) # 0 f(@ (-95 for critical number f'(x) l'(a) f'(a) = 0 (29)" - 0 d=9 f") 20 (4-9) Alo f@ f") = 60 (2-9), f'(a) = 120(249) f'(a) = {"(9)=1"Ő-FW) (q) = 0 Na = 20 120 #0 since n=5 no Hence f(x) = @-9) has extremum at 2=9 Note: Let f: I -> R and c be an interior Point of I. If f'(c) = f" (C) = fW-(C) = 0 and (M) (c) 0, then t han 6 no extremum at a=c if n be odd. and ☺ a local extremum at x=c if n be even. a local minimum if fm (c)7.0, a local maximum it tm) CC) < А Ans: The critical number is at x= There is no local minimum and no local maximum.

1600 x f (2) 21 _ 8002 f'() 473 "For critical Point Point f'(2) 473-1600 x (2x - 1600) -0. 4x - 16000 2v + 20 Critical number are 0, 20, -20 f"(a) = 12 -1600 Lo At 2 = 0 f"(0) - 16000 since f'(o) = 0 & f "(0) To , f has local maximum at a=0. At a= 20 f"(20) 12 » (20) ~- 1600 4800 - 16.00 3200 > 0 since t' (20) = 0 and t" (20) > 0 f(x) has local minimum ant x=20. At f" (-20) = 12 * (-20)-1600 3200 yo since fo(-20) -o and f"(-20) 70 a f(a) has local minimum at x=-20 2 = -20

Ans' © The critical numbers are at O, -20, 20 There is There is a local maximum at a=0 local minimum at x = +20. 2 x 74 y for critical 0 x- 2-1 Y (2-1).27 fe-7). 1 x-4) ~ 2x~ - 8x -x + 7 2-4) x - 8x + (-) numbers of 7 Y'() x- 8x+ (2-4 ~ x- 8x + 7 6 - 1)(-7) = 0 a =), (2-1)".622-8) - (27 82+7). 2(2-4) Y"(~) (1-4) A 26x-) { (-1)^- 24+81-3) a - A) 9 2 (x^-82 +16 -x+82-7) 2-13 O af a 다. 0. x=7

18 29 G-a) 6-1) Now at 2=1 18 18 CV/M <0 7" 27 (1-1) and y" (1) since Y' O) Y (1) han local maximum at xa1 Now at x= 7 2 - 27 18 Y"(7) 18 7-4)3 4' (7)=0 y (2) has local minimum at a=7 since and y" (7) 70. The local Ans! A and the maxima occur at a=1 local minima occur at a=7