# How do you find all points of inflection given y=-(4x+20)^(1/3)?

Jul 7, 2017

No points of inflection.

#### Explanation:

To find points of inflection, we find the second derivative of the function and set it equal to zero. We can then find the POI from the values that will make the equation true, i.e. equal to zero.

$y = - {\left(4 x + 20\right)}^{\frac{1}{3}}$

$\implies y ' = - \frac{4}{3} {\left(4 x + 20\right)}^{- \frac{2}{3}}$

$\implies y ' ' = \frac{32}{9} {\left(4 x + 20\right)}^{- \frac{5}{3}}$

We set the second derivative equal to zero...

$\frac{32}{9} {\left(4 x + 20\right)}^{- \frac{5}{3}} = 0$

Which is equivalent to:

$0 = \frac{32}{9 {\left(4 x + 20\right)}^{\frac{5}{3}}}$

And we can see that no matter what we do, we will never find a value of x that will make this true! Therefore, there are no points of inflection.