# What are the critical values, if any, of f(x)= x^2*(1 + 3 ln x)?

Sep 7, 2017

To find critical values, we just have to set $f ' \left(x\right) = 0$

#### Explanation:

$f \left(x\right) = {x}^{2} \cdot \left(1 + 3 \ln x\right)$

$\implies f ' \left(x\right) = 2 x \cdot \left(1 + 3 \ln x\right) + \left({x}^{2}\right) \cdot \left(\frac{3}{x}\right)$
$\implies f ' \left(x\right) = 2 x \cdot \left(1 + 3 \ln x\right) + 3 x$
$\implies f ' \left(x\right) = 0$
$\implies 2 x \cdot \left(1 + 3 \ln x\right) + 3 x = 0$
$\implies x \left(2 \cdot \left(1 + 3 \ln x\right) + 3\right) = 0$
$\implies x = 0$ or $\left(2 \cdot \left(1 + 3 \ln x\right) + 3\right) = 0$
$\implies x = 0$ or $x = {e}^{- \frac{5}{6}}$