Question

Statement-I : 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 $3^{1/3}$.
Because :
Statement-II : $x^{1/x}$ is increasing for $0<x<e$ and decreasing for $x>e$.

• A. Statement-I is true, Statement-II is true ; Statement-II is correct explanation for Statement-I.
• B. Statement-I is true, Statement-II is true ; Statement-II is NOT a correct explanation for statement-I.
• C. Statement-I is true, Statement-II is false.
• D. Statement-I is false, Statement-II is true.

Answer 1

Answer: (A)

Consider $f(x)=x^{1/x}$
$f^{\prime }(x)=x^{1/x}\left(\frac{1-\ell nx}{x^2}\right)\forall x>0$

$\therefore$ at $x=e,f(x)$ has absolute maximum value.
$3^{1/3}>4^{1/4}=2^{1/2}$.
Hence both statements are true statement-II explains statements I.