Question

How do you create a cubic polynomial function with a local max of 2, local min of -2 , and a point of inflection at 0?

Answer 1

suppose y = ax^3+bx^2+cx+d
so, y" = 6ax+2b
you know that
y"(0) = 0
so, b=0
y" = 6ax
since y(2) is a maximum, y"(2) < 0, so a<0
y' = m(x-2)(x+2) = m(x^2-4) = 3ax^2+c
y'(2)=y'(-2)=0, so 12a+c = 0
c = -12a
so,
y = ax^3-12ax+d
so, let's try some value for a, say -1. Then we have
y = -x^3+12x
The d does not matter.
You can see from the graph that we are done.

https://www.wolframalpha.com/input/?i=-x%5E3%2B12x