Question

What are the inflection points for `x^3 + 5x^2 + 4x - 3`?

Answer 1

An inflection point is defined when a function is changing from concave to convex (or vice-versa), that is, changing its concavity.

Thus, the second derivative:

`(dy)/(dx)=3x^2+10x+4`

`(d^2y)/(dx^2)=6x+10`

Equaling the second derivative to zero (because at the inflection point, the slope's not changing, it's the very point where concavity changes, thus being of derivative equal to zero:

`6x+10=0`
`x=-10/6=-5/3~=-1.67`

Substituting this `x` coordinate in the original function, in order to get the `y` coordinate:

`y=(-5/3)^3+5(-5/3)^2+4(-5/3)-3`
`y=(-125/27)+(125/9)-20/3-3`
`y=(-125+375-180-81)/27`
`y=-11/9~=-1.22`

So, you inflection point is `(-1.67,-1.22)`

graph{x^3+5x^2+4x-3 [-10, 10, -5, 5]}