Differentiation and its Application: If f(x)=xe^x(1-x) , then f(x) is | IIT-JEE 2001 Screening

IIT-JEE 2001 ScreeningQuestion 1

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

Differentiation and its Application

increasing on $[-1/2,1]$✓ Correct answerdecreasing on $R$increasing on $R$decreasing on $[-1/2,1]$

Solution

Concept. Factor the derivative and read its sign. Formulas used. (1) $\dfrac{d}{dx}e^{x(1-x)}=(1-2x)e^{x(1-x)}$Step 1 - Derivative. By (1), $$f'(x)=e^{x(1-x)}\big(1+x-2x^2\big)=e^{x(1-x)}(1-x)(1+2x).$$ Step 2 - Sign. Positive for $-\frac12<x<1$. So $f$ increases on $\left[-\frac12,1\right]$. Answer: (A)

Differentiation and its Application - JEE Mathematics question (IIT-JEE 2001 Screening)

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