# What are the points of inflection, if any, of f(x)=x^(1/3)e^(3x) ?

Nov 2, 2016

there are two of them and exactly they are $x = \setminus \frac{- 1 - \sqrt{3}}{9}$ and $x = \setminus \frac{- 1 + \sqrt{3}}{9}$

#### Explanation:

It is matter of deriving twice $f \left(x\right)$
$f ' \left(x\right) = {e}^{3 x} \left(\setminus \frac{1}{3 \sqrt[3]{{x}^{2}}} + 3 \sqrt[3]{x}\right)$
that after a simple algebraic manipulation yelds
$f ' \left(x\right) = {e}^{3 x} \left(\setminus \frac{1 + 9 x}{3 \sqrt[3]{{x}^{2}}}\right)$
so that the point of minimum has got abscissa $x = - \setminus \frac{1}{9}$
By deriving $f ' \left(x\right)$ we obtain
$f ' ' \left(x\right) = \setminus \frac{- 2}{9 \sqrt[3]{{x}^{5}}} + \setminus \frac{2}{\sqrt[3]{{x}^{2}}} + 9 \sqrt[3]{x}$
that after a simple manipulation
turns to be
$f ' ' \left(x\right) = \setminus \frac{- 2 + 18 x + 81 {x}^{2}}{9 \sqrt[3]{{x}^{5}}}$
whose roots are just the abscissas of the inflection points