# What are the points of inflections for f(x) = (x^2) - 3/x^3?

Jan 19, 2016

Point of inflection is $x \approx 1.783$

#### Explanation:

To find points of inflection of a function $f \left(x\right)$ we need to find second derivative of $f \left(x\right)$ and equate it to zero. Then find the roots of the polynomial thus obtained.
$f \left(x\right) = \left({x}^{2}\right) - \frac{3}{x} ^ 3$
or $f \left(x\right) = {x}^{2} - 3 {x}^{-} 3$

First derivative $f ' \left(x\right) = 2 x + 9 {x}^{-} 4$
Second derivative $f ' ' \left(x\right) = 2 - 36 {x}^{-} 5$
For point of inflection $f ' ' \left(x\right) = 0$
$\implies$ $2 - 36 {x}^{-} 5 = 0$
$\implies 36 {x}^{-} 5 = 2$

Solving for $x$
Only root is $x = \sqrt[5]{\frac{36}{2}}$