Local max, min and point of inflection of $y= ax^3 + 3bx^2 + 3cx - d$

Question

Suppose $$y= ax^3 + 3bx^2 + 3cx - d$$ has a local maximum at $P_1(x_1, y_1)$ and local minimum at $P_2(x_2, y_2)$. Prove that the point of inflection is the midpoint of $P_1P_2$.

I would mention what I've already done till now in regard to this question, but I really have no clue how to do it. If someone does know how to do it, please help me out.

Thank you.

Answer 1

HINTS:

1. What is the derivative of this function? What is the second derivative?

2. If a function is differentiable and has a local max/min, what can you say about the derivative?. Does this give you any equations in $x$?

3. If you know something has a point of inflection, what can you say about a functions second derivative? Doesn't this also give you an equation in $x$?

4. Can you use the equations from 2 to see the equation from 3 has solution $x$ being the midpoint (average) of $x_1,x_2$?

5. At this inflection value for $x$, what is the $y$ value? Does it happen to be the midpoint (average) of the $y$'s?