q0500771493 發問時間: 科學數學 · 4 年前

數學 For the function f(x)=3√x(x-7)2 which of the following statement are true?

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For the function f(x)=3√x(x-7)2 which of the following statement are true?

A f (7) = 0 is a relative minimum

B f (x) is differentiable everywhere

C f (x) has no relative maximum

D f (0) = 0 is a relative minimum

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    Question:

    For the function f(x) = (∛x) (x - 7)², which of the following statement are true?

    A. f(7) = 0 is a relative minimum

    B. f(x) is differentiable everywhere

    C. f(x) has no relative maximum

    D. f(0) = 0 is a relative minimun

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    Solution:

    f(x) = (∛x) (x - 7)²

    f'(x)

    = [ (∛x) ]' (x - 7)² + (∛x) [ (x - 7)² ]'

    = 1/[3(∛x)²] (x - 7)² + (∛x) [ 2(x - 7) ]

    = (x - 7)²/[3(∛x)²] + 2 (∛x) (x - 7)

    f'(x) = 0

    (x - 7)²/[3(∛x)²] + 2 (∛x) (x - 7) = 0

    (x - 7) { (x - 7)/[3(∛x)²] + 2 (∛x) } = 0

    (x - 7)/[3(∛x)²] + 2 (∛x) = 0   or   x - 7 = 0

    x - 7 = - 6 (∛x)³        or   x = 7

    x + 6x = 7           or   x = 7

    x = 1              or   x = 7

    f'(x) = (x - 7)²/[3(∛x)²] + 2 (∛x) (x - 7)

    f'(x) > 0 when x < 1

    f'(x) < 0 when 1 < x < 7

    f'(x) > 0 when x > 7

    ∵ f'(x) < 0 when 1 < x < 7

     f'(x) > 0 when x > 7

    ∴ A. f(7) = 0 is a relative minimum

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    For B, C, D,

    If x = 0, f'(0) is undefined.

    f(x) is not differentiable at x = 0

    ∴ B. is incorrect.

    ∵ f'(x) > 0 when x < 1

     f'(x) < 0 when 1 < x < 7

    Then, f(1) = 0 is a relative maximum.

    ∴ C. is incorrect.

    f'(x) > 0 when x < 0

    f'(x) > 0 when 0 < x < 1

    ∴ D. is incorrect.

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    The First Derivative Test

    Suppose f(x) is a differentiable function.

    If f'(x) changes from negative to positive at c, then f(c) is a relative minimum ( a local minimum )

    If f'(x) changes from positive to negative at c, then f(c) is a relative maximum ( a local maximum )

    If f'(x) does not change at c, then f(c) is not a relative extremum ( a local extremum )

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