Question

The interval in which the function $f(x) = x^x, \, x>0$, is strictly increasing is:

• A. $\left( 0, \frac{1}{e} \right]$
• B. $\left[ \frac{1}{e^2}, 1 \right)$
• C. $(0, \infty)$
• D. $\left[ \frac{1}{e}, \infty \right)$

Answer 1

Answer: (D)

$\left[ \frac{1}{e}, \infty \right)$

Answer 2

Given:

f(x) = x^x, \quad x > 0\.

Taking the natural logarithm:

$f(x) = x \ln x.$

Differentiating:

$\frac{1}{y} \frac{dy}{dx} = \ln x + 1 \implies \frac{dy}{dx} = x^x(1 + \ln x).$

For $f(x)$ to be strictly increasing:

\frac{dy}{dx} > 0 \implies 1 + \ln x > 0\.

Solve:

$\ln x > -1 \implies x > \frac{1}{e}.$

Thus, the function is strictly increasing in:

$\left[\frac{1}{e}, \infty\right).$