# What are the points of inflection, if any, of f(x)=x^2 - 3/x^3 ?

Dec 15, 2016

$\left({18}^{\frac{1}{5}} , {18}^{\frac{2}{5}} - \frac{3}{18} ^ \left(\frac{3}{5}\right)\right) = \left(1.783 , 2.648\right)$, nearly.. Look for this point in the inserted graph, at which the tangent is crossing the curve.

#### Explanation:

At a point of inflexion, y'' = 0

graph{x^2-3/x^3 [-2, 5, -5, 5]}

$y = {x}^{2} - \frac{3}{x} ^ 3$

$y ' = 2 x + \frac{9}{x} ^ 4$

y''=2-36/x^5=0, at x =18^(1/5

Further, if y''' is not 0 here, this gives a point of inflexion.

Here, y''' is not 0.

So, (18^(1/5), 18^(2/5-3/18^(3/5)=(1.783, 2.648), nearly, is the point of

inflexion (POI).

The scale on the x-axis is changed to disclose tangent-crossing-

curve at POI.s