Question

Assertion :The greatest of the numbers $1,2^{1/2},3^{1/3},4^{1/4},5^{1/5},6^{1/6},7^{1/7},is{3}^{1/3}.$ Reason: $x^{1/x}$ is increasing for $0<x<e$ and decreasing for x > e

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Assertion is incorrect but Reason is correct

Answer 1

The correct option is A Both Assertion and Reason are correct and Reason is the correct explanation for Assertion

Given, $f\left(x\right)=x^{1/x}$
$\ln f\left(x\right)=\frac{1}{x}\ln x$

$\frac{1}{f\left(x\right)}{f}^{{}^{\prime }}\left(x\right)=\frac{1}{x^2}-\frac{\ln x}{x^2}$

$f^{{}^{\prime }}\left(x\right)=f\left(x\right)(\frac{1}{x^2}-\frac{\ln x}{x^2})$

$f^{{}^{\prime }}\left(x\right)=x^{1/x}(\frac{1}{x^2}-\frac{\ln x}{x^2})$
for $xϵ$ (0,e) $f^{{}^{\prime }}\left(x\right)is>0$. it is increasing function in this interval and decreasing afterwards.

At $x=e$ function has maxima.
$3^{1/3}>4^{1/4}=2^{1/2}.$

Thus both the assertion and reason are correct and the assertion is followed by the reason.