JEE MAIN - Mathematics (2024 - 6th April Morning Shift - No. 19)
The interval in which the function $$f(x)=x^x, x>0$$, is strictly increasing is
$$(0, \infty)$$
$$\left(0, \frac{1}{e}\right]$$
$$\left[\frac{1}{e^2}, 1\right)$$
$$\left[\frac{1}{e}, \infty\right)$$
Explanation
$$\begin{aligned}
& f(x)=x^x \\
& f(x)=x^x(\log x+1) \\
& f(x) \geq 0 \\
& \Rightarrow 1+\log x \geq 0 \\
& \Rightarrow \log x \geq-1 \\
& \Rightarrow x \geq e^{-1} \\
& \therefore x \in\left[\frac{1}{e^{\prime}}, \infty\right)
\end{aligned}$$
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