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If f(x)=x e^(x(x-1)) , then f(x) is incr...

If `f(x)=x e^(x(x-1))` , then `f(x)` is increasing on `[-1/2,1]` decreasing on `R` increasing on `R` (d) `` decreasing on `[-1/2,1]`

A

increasing in `[-1//2, 1]`

B

decreasing in R

C

increasing in R

D

decreasing in `[-1//2, 1]`

Text Solution

Verified by Experts

The correct Answer is:
B
Given, ` f(x) = x e ^(x ( 1 - x))`
`rArr f ' (x) = e ^(x(1- x)) + x e (x (1- x)) (1 - 2x ) `
`" " = e ^(x(1- x)) [ 1 + x (1 - 2x )]`
`" " = e ^(x ( 1 - x )) ( 1+ x - 2x ^(2))`
`" " = - e^(x (1 - x )) ( 2x^(2) - x - 1)`
` " " = - e^(x( 1-x )) (x -1) (2 x + 1)`
Which is positive in `(-(1)/(2), 1)`
Therefore, `f(x)` is increasing in `[- (1)/(2), 1]`
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