The explicit problem is: if a real function $f(x)$ has the continuous third-order derivative over $\mathbb{R}$ then $\exists\,\zeta\in\mathbb{R}$ s.t. $f(\zeta)f'(\zeta)f''(\zeta)f'''(\zeta)\geq0$.
If $f(x),f'(x),f''(x)$ and $f'''(x)$ have zeros over $\mathbb{R}$ then it is proved. So suppose $f(x)>0$ identically. But I have no ideas about the latter part of the proof, could you give me some hints?