In the interval `[0,1],` the function `x^(25)(1-x)^(75)` takes its maximum value at the point (a) 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

Let `f(x)=x^(25)(1-x)^(75)` Then
`f(x) =x^(24)(1-x)^(74)(1-4x)`
Clearly f'`(x) gt 0` in the left neighborhood of 1/4 and `f(x) lt 0 ` in the right neighbourhood of 1/4 So, f(x) changes its sign form positive to negative in the nieghbourhood of 1/4.
Hence , it attains maximum at x= 1/4.