For the function $f(x) = x^x,x > 0$, which of the following are TRUE?

(A) $f'(x) = x^x(1+ \log x)$
(B) $x=e$ is the critical point
(C) $f$ is increasing in $(\frac{1}{e},∞)$
(D) $f$ is increasing in $(0, ∞)$

Choose the correct answer from the options given below:

(A) and (C) only

The correct answer is Option (3) → (A) and (C) only

Given $f(x) = x^{x},\; x > 0$

$\ln f = x \ln x \Rightarrow \frac{f'(x)}{f(x)} = \ln x + 1$

$\Rightarrow f'(x) = x^{x}(1 + \ln x)$

⟹ (A) is True.

For critical points, $f'(x) = 0 \Rightarrow 1 + \ln x = 0 \Rightarrow x = \frac{1}{e}$

⟹ (B) is False (critical point is at $x = \frac{1}{e}$, not $x = e$).

Sign of $f'(x)$:

$f'(x) > 0$ for $x > \frac{1}{e}$ and $f'(x) < 0$ for $0 < x < \frac{1}{e}$

⟹ $f(x)$ is increasing on $\left(\frac{1}{e}, \infty\right)$

⟹ (C) is True, (D) is False.

Correct statements: (A) and (C)