Two statements are given, one labelled Assertion (A) and the other labelled Reason (R).

Assertion (A): The function $f (x) = x^x, x > 0$ is strictly increasing in $[\frac{1}{e},∞)$.
Reason (R): $\log_a, x>b⇒x> a^b$ if $a > 1$.

Select the correct answer from the options given below.

Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of the Assertion (A).

The correct answer is Option (2) → Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of the Assertion (A).

$f(x) = x^x$

Taking logarithm on both sides, we get

$\log f (x) = x \log x$

Differentiating w.r.t. x, we get

$\frac{1}{f(x)}.f'(x)=\frac{x}{x}+\log x$

$⇒ f'(x) = x^x(1+ \log x)$

For function to be strictly increasing $f'(x) > 0$

i.e. $x^x (1+\log x)>0⇒1+ \log x>0$  $(∵ x^x>0)$

$⇒\log x>-1⇒x>e^{-1}$  $(∵ e> 1)$

$⇒x>\frac{1}{e}$

∴ f(x) is strictly increasing in $[\frac{1}{e},∞)$

∴ Assertion is true.

Also, $\log_a, x>b⇒ x> a^b$ if $a > 1$ is true

∴  Reason is true.

Hence, Assertion and Reason both are true and Reason is the correct explanation of Assertion.