Problem #PRU-111264

Problems Calculus Functions of one variable. Continuity Continuity and compactness Monotonicity, boundedness

Problem

A continuous function \(f(x)\) is such that for all real \(x\) the following inequality holds: \(f(x^2) - (f (x))^2 \geq 1/4\). Is it true that the function \(f(x)\) necessarily has an extreme point?