Is there a simple reason for why the two stationary points in a cubic polynomial has its second derivative equal in magnitude but opposite in sign.

Question

Given a polynomial $f(x) = x^3 + ax^2 + bx + c$, then
$$f'(x) = 3x^2 + 2ax + b$$ $$f''(x) = 6x + 2a$$
The stationary points in the curve will have $x = \frac{-a \pm \sqrt{a^2 - 3b}}{6}$, but then the two points have their second derivative equal in magnitude but opposite in sign since
$$f''(\frac{-a + \sqrt{a^2 - 3b}}{3}) = \sqrt{a^2 - 3b} = -f''(\frac{-a - \sqrt{a^2 - 3b}}{3})$$
I remembered that I once had an intuition why this is true but I forgot now. Could somebody tell me why? Thanks in advance

Answer 1

If you translate the graph so that the single point of inflection is the origin then the resulting function is odd.