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In the interval [0,1], the function x^(2...

In the interval `[0,1],` the function `x^(25)(1-x)^(75)` takes its maximum value at the point 0 (b) `1/4` (c) `1/2` (d) `1/3`

A

0

B

`1//4`

C

`1//2`

D

`1//3`

Text Solution

Verified by Experts

The correct Answer is:
B
Let ` f(x) = x ^( 25) ( 1- x ) ^( 75), x in [ 0, 1]`
`rArr f ' (x) = 25 x ^( 24) ( 1 -x ) ^(75) - 75 x ^( 25) (1 - x ) ^(74) `
`" " = 25 x ^( 24) ( 1- x ) ^( 74) [ ( 1- x ) - 3x]`
`" " = 25 x ^( 24 ) (1 - x ) ^( 74) (1 - 4x)`
For maximum value of `f(x)`, put `f' ( x) =0`
`rArr " " 25 x^( 24) (1- x ) ^( 74) ( 1- 4x)`
`rArr " " x = 0, 1, (1)/(4)`
Also, at ` " " x = 0 , y = 0`
At ` " " x = 1, y = 0`
and at ` x = 1//4, y gt 0`
`therefore f(x)` attains maximum at ` x = 1//4`
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