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exists x_o in R: F(x_0) F'''(x_0)>=0 (VII Soros Olympiad 2000-01 R1 11.3)

Source:

July 29, 2021
analysisalgebraderivative

Problem Statement

The function F(x)F (x) is defined on RR and has a second derivative for each value of the variable. Prove that there is a point x0x_0 such that the product F(x0)F′′(x0) F(x_0) F''(x_0) is non-negative.
PS. In my [url=http://www.1543.su/olympiads/soros/20002001/1/1soros00.htm]source, it is not clear if it means F(x0)F′′(x0) F(x_0) F''(x_0) or F(x0)F′(x0) F(x_0) F'(x_0).
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