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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Answer:

(b)

Solution:

  1. 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 $-\frac{1}{2}, 1$.



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