Observe the statements given below : Assertion (A) : $f(x)=x e^{-x}$ has the maximum at $x=1$ Reason (R) :…
Observe the statements given below :
Assertion (A) : $f(x)=x e^{-x}$ has the maximum at $x=1$
Reason (R) : $f^{\prime}(1)=0$ and $f^{\prime \prime}(1) < 0$
Which of the following is correct?
Both $(A)$ and (R) are true and (R) is the correct reason for (A)
Both $(A)$ and $(R)$ are true, but (R) is not the correct reason for (A)
(A) is true, (R) is false
(A) is false, (R) is true
Solution
Given, $f(x)=x e^{-x}$
$\begin{aligned} f^{\prime}(x) & =e^{-x}-x e^{-x} \\ f^{\prime \prime}(x) & =-e^{-x}-e^{-x}+x e^{-x} \\ & =-2 e^{-x}+x e^{-x}\end{aligned}$
For maximum, put
$f^{\prime}(1)=0 \Rightarrow x=1$ and $f^{\prime \prime}(1)=-1 < 0$
$\therefore$ Both $\mathrm{A}$ and $\mathrm{R}$ are true and $\mathrm{R}$ is the correct reason for $\mathrm{A}$.