Applications of Derivatives
Monotonicity of $x^x$
nta_pyq_2024_apr
Grade 12
Question:
The interval in which the function $f(x)=x^x$, $x>0$, is strictly increasing is:
$\left(0,\dfrac{1}{e}\right]$
$(0,\infty)$
$\left[\dfrac{1}{e},\infty\right)$
$\left[\dfrac{1}{e^2},1\right)$
Step-by-Step Solution
Key Concept: $f'(x)=x^x(1+\ln x)>0\Leftrightarrow x>1/e$.
$f'(x)>0\Leftrightarrow x\geq1/e$.
Correct Answer: 3