Application of Derivatives 2 Question 10
10. If $f(x)=x e^{x(1-x)}$, then $f(x)$ is
(2001, 2M)
(a) increasing in $[-1 / 2,1]$
(b) decreasing in $R$
(c) increasing in $R$
(d) decreasing in $[-1 / 2,1]$
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Solution:
- Given, $f(x)=x e^{x(1-x)}$
$$ \begin{aligned} \Rightarrow \quad f^{\prime}(x) & =e^{x(1-x)}+x e^{x(1-x)}(1-2 x) \\ & =e^{x(1-x)}[1+x(1-2 x)] \\ & =e^{x(1-x)}\left(1+x-2 x^{2}\right) \\ & =-e^{x(1-x)}\left(2 x^{2}-x-1\right) \\ & =-e^{x(1-x)}(x-1)(2 x+1) \end{aligned} $$
which is positive in $-\frac{1}{2}, 1$.
Therefore, $f(x)$ is increasing in $-\frac{1}{2}, 1$.
