Question

Observe the statements given below :
Assertion (A) : $f(x)=x e^{-x}$ has the maximum at $x=1$
Reason (R) : $f'(1)=0$ and $f''(1)<0$
Which of the following is correct?

• A. Both (A) and (R) are true and (R) is the correct reason for (A)
• B. Both (A) and (R) are true, but (R) is not the correct reason for (A)
• C. (A) is true, (R) is false
• D. (A) is false, (R) is true

Answer 1

Answer: (A)

Given, $f(x) =x e^{-x}$
$f'(x) =e^{-x}-x e^{-x}$
$f''(x) =-e^{-x}-e^{-x}+x e^{-x}$
$=-2 e^{-x}+x e^{-x}$
For maximum, put
$f'(1)=0 \Rightarrow x=1$ and $f''(1)=-1<0$
$\therefore $ Both $A$ and $R$ are true and $R$ is the correct reason for $A$.