What are the points of inflection, if any, of f(x) = 4x^6 +4x^5 -5x^4 ?

Apr 23, 2016

$x = - 1.155 , .448$

Explanation:

Let's find the second derivative...

$f ' \left(x\right) = 24 {x}^{5} + 20 {x}^{4} - 20 {x}^{3}$

$f ' ' \left(x\right) = 120 {x}^{4} + 80 {x}^{3} - 60 {x}^{2}$

There it is, the second derivative. How about we set the second derivative equal to zero...

$f ' ' \left(x\right) = 120 {x}^{4} + 80 {x}^{3} - 60 {x}^{2} = 0$

$20 {x}^{2} \left(6 {x}^{2} + 4 x - 3\right) = 0$

$x = - 1.155 , 0 , .448$

Do some analysis of these points, just to make sure. You'll be glad you did that, for your answers are...

$x = - 1.155 , .448$