You're right. The more I'm thinking about it, the more I believe the statement is true. I'll try to prove it but I'm not sure I'll success, so I hope someone else will do it. Anyway it's an interesting problem.
I might miss something here but I do see your example proving that my statement is just correct.
Plot f(x) = x^8 + 1 and f"(x) = 56*x^6.
I see that in any interval (a,b) both f(x) and f"(x) reach max or min at the same ordinate.
I tried defeating my statement with exponentials, logarithms, trigonometrics functions, basic polinomials.
It always came out clean and valid.
I'd really need to see a function that proves the contrary.