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.