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:
Answer & explanation
Correct answer: option 3
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)