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`

Options

A. 0
B. `1//4`
C. `1//2`
D. `1//3`

Answer

B

Explanation

Given, `f(x)=x^(25)(1-x)^(75)`
`Rightarrow f'(x)=25x^(24)(1-x)^(75)-75x^(25)(1-x)^(74)`
`=25x^(24)(1-x)^(74)(1-4x)`
`therefore f'(x)=0`
`Rightarrow x=0,1,1//4`
`"If "x lt 1//4,"then "`
`f(x)=25x^(24)(1-x)^(74)(1-4x)gt 0`
and if `x gt 1//4`, then
`f'(x)=25x^(24)(1-x)^(74)(1-4x) lt 0`
Thus, f'(x) changes its sign from positive to negative as x passes through 1//4 from left to right.
Hence, f(x) attains its maximum at x=1/4.