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Mathematics Question on Application of derivatives

If f(x)=xex(1−x)f (x) = xe^{x(1-x)} , then f (x) is

A

increasing on [-1/2, 1]

B

decreasing on R

C

increasing on R

D

decreasing on [-1/2, 1]

Answer

increasing on [-1/2, 1]

Explanation

Solution

The correct option is(A): increasing on [-1/2, 1]

f(x)=xex(1−x)f (x) = xe^{x (1 - x)}
⇒ f′(x)=ex(1−x)+(1−2x)xex(1−x)\Rightarrow \, f'(x) = e^{x (1-x)} + (1 - 2x) x e^{x (1-x)}
=−ex(1−x)(2x2−x−1)=−ex(1−x)(2x+1)(x−1)= - e^{x (1-x) } (2x^2 - x - 1) = - e^{x (1 - x) } (2x + 1) (x - 1)
∴\therefore f (x) is increasing on [-1/2, 1]