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Q.
If $f\left(x\right) = xe^{x\left(1-x\right)}, x \in R$ , then $f\left(x\right)$ is
AIEEEAIEEE 2012Application of Derivatives
Solution:
$ f\left(x\right) = xe^{x\left(1-x\right)}, x \in R $
$f'\left(x\right) = e^{x\left(1-x\right)}. \left[1+x-2x^{2}\right]$
$= -e^{x\left(1-x\right)}\left[2x^{2}-x-1\right]$
$= -2x^{x\left(1-x\right)}\left[\left(x+\frac{1}{2}\right)\left(x-1\right)\right]$
$f'\left(x\right) = -2e^{x\left(1-x\right)}.A$
where A $= \left(x+\frac{1}{2}\right)\left(x-1\right)$
Now, exponential function is always +ve and $f'\left(x\right)$ will be opposite to the sign of A
which is -ve in $\left[-\frac{1}{2}, 1\right]$
Hence, $f'\left(x\right)$ is +ve in $\left[-\frac{1}{2},1\right]$
$\therefore f\left(x\right)$ is increasing on $\left[-\frac{1}{2},1\right]$