Quote:

Isaac,

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.

Also,

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.

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.