(a) Find the critical numbers of the function f(x) = x^8(x ? 2)^7. x = ________?
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Question
(a) Find the critical numbers of the function f(x) = x^8(x ? 2)^7. x = ________? (smallest value) x = ________? x =_________? (largest value) (b) What does the Second Derivative Test tell you about the behavior of f at these critical numbers? At x =___________? , the function has ---Select--- a local minimum a local maximum neither a minimum nor a maximum. (c) What does the First Derivative Test tell you that the Second Derivative test does not? (Enter your answers from smallest to largest x value.) At x =____________? , the function has ---Select--- a local minimum a local maximum neither a minimum nor a maximum. At x =____________? , the function has ---Select--- a local minimum a local maximum neither a minimum nor a maximum.Explanation / Answer
f(x) = (x^8)(x - 2)^7
f'(x) = 8(x^7)(x - 2)^7+ 7(x^8)(x - 2)^6
. . . = [(x^7)(x - 2)^6][8(x - 2) + 7x]
. . . = (x^7)(15x - 16)(x - 2)^6
The critical points are x = 0, x = 16/15 and x = 2
For ?? < x < 0: x^7 < 0, 15x - 16 < 0 and (x - 2)^6 > 0,
therefore f'(x) > 0 and f(x) is increasing
For 0<x<16/15: x^7 > 0, 15x - 16 < 0 and (x - 2)^6 > 0,
therefore f'(x) < 0 and f(x) is decreasing.
For 16/15<x<2: x^7 > 0, 15x - 16 > 0 and (x - 2)^6 > 0,
therefore f'(x) > 0 and f(x) is increasing.
For 2 < x <+? : x^7 > 0, 15x - 16 > 0 and (x - 2)^6 > 0,
therefore f'(x) > 0 and f(x) is increasing.
f(x) gets a local maximum at x = 0, a local minimum at x = 16/15
and an inflexion at x = 2
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