Question

The function $f(x)=xe^{x(1-x)}$,x∈R is

• A. increasing in $(-\frac{1}{2} , 1)$
• B. increasing in $(-\frac{1}{2} , 1)$
• C. increasing in $(-1 , -\frac{1}{2})$
• D. decreasing in $(-\frac{1}{2} , \frac{1}{2})$

Answer 1

Answer: (A)

increasing in $(-\frac{1}{2} , 1)$

Answer 2

$f(x)=xe^{x(x−1)}$

⇒ $f′(x)=e^{x(1−x)}+x^2(−1)e^{x(1−x)}+x(1−x)e^{x(1−x)}$

=−$e^{x(1−x)}(2x^2−x−1)=−e^{x(1−x)}(x−1)(x+1/2)$

$f′(x)=0$ at x=−1/2$$,1 and $f′(x)>0 \space when −1/2<x<1$

⇒ f(x) is increasing function on [−1/2,1]