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.
Answer & explanation
Correct answer: option 1
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.