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

1 Answer
May 25, 2015

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]}