What are the points of inflection, if any, of f(x)= x^3 + 9x^2 + 15x - 25 ?

Dec 6, 2015

$\left(- 3 , - 70\right)$

Explanation:

Inflection points (change in concavity) occur when the second derivative is zero.

$f ' \left(x\right) = 3 {x}^{2} + 18 x + 15$

$f ' ' \left(x\right) = 6 x + 18$

Clearly $f ' ' \left(x\right) = 0 \iff x = - 3$

But $f \left(- 3\right) = - 27 + 27 - 45 - 25 = - 70$

Therefore the point $\left(- 3 , - 70\right)$ is an inflection point.

This can also be viewed from the graph of this cubic polynomial function :

graph{x^3+9x^2+15x-25 [-83.3, 83.34, -41.66, 41.63]}