# What are the critical numbers of f(x) = x^2*(1 + 3 ln x)?

The only critical number is $x = {e}^{- \frac{5}{6}}$
$f ' \left(x\right) = 2 x \left(1 + 3 \ln x\right) + {x}^{2} \left(\frac{3}{x}\right) = x \left(6 \ln x + 5\right)$
$f ' \left(x\right)$ is defined for all $x$ in the domain of $f$ and
$f ' \left(x\right) = 0$ at $\ln x = - \frac{5}{6}$ so $x = {e}^{- \frac{5}{6}}$.