1. Understand the problem:
We need to determine where the function \( f(x) = x^x \) (for \( x > 0 \)) is strictly increasing.
2. Use logarithmic differentiation:
Let \( y = x^x \). Take the natural logarithm:
\[ \ln y = x \ln x \]
Differentiate implicitly with respect to \( x \):
\[ \frac{1}{y} \frac{dy}{dx} = \ln x + 1 \]
Thus:
\[ f'(x) = y (\ln x + 1) = x^x (\ln x + 1) \]
3. Determine where \( f'(x) > 0 \):
Since \( x^x > 0 \) for \( x > 0 \), the sign of \( f'(x) \) depends on \( \ln x + 1 \):
\[ \ln x + 1 > 0 \implies \ln x > -1 \implies x > e^{-1} = \frac{1}{e} \]
Correct Answer: (C) \( x > \frac{1}{e} \)
Rewrite $y = x^x$ as $y = e^{x \ln x}$. Differentiate: \[ \frac{dy}{dx} = y (\ln x + 1). \] The sign of $\ln x + 1$ determines monotonicity. Solve $\ln x + 1 > 0$: \[ \ln x > -1 \implies x > \frac{1}{e}. \] Thus, $x^x$ is strictly increasing for $x > \frac{1}{e}$.