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?

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

### 1 個解答

• 最佳解答

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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