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Question

On the interval [0,1], the function x25(1-x)75 takes its maximum value at the point


A

0

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B

12

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C

13

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D

14

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Solution

The correct option is D

14


Explanation for the correct option.

Finding maximum value of the given function

Given : f(x)=x25(1-x)75

Differentiating with respect to .x
f'(x)=25x24(1x)7575x25(1x)74=25x24(1x)74(14x)f'(x)=0x=0,1,14

Ifx<14, then
f(x)=25x24(1x)74(14x)>0
and ifx>14​,then
f(x)=25x24(1x)74(14x)<0

Thus f(x)changes its sign from positive to negative as x passes through 14 from left to right.
Hence f(x) attains its maximum at x=14

Hence the correct answer is option (D).


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