Derivatives and Solutions

This lecture discusses a function F defined on real numbers, assuming F is twice differentiable and its second derivative is continuous. It explores four propositions and determines which are false by examining each one individually. The analysis includes applying the mean value theorem to show the existence of points where the second derivative of F vanishes. Additionally, it constructs a counterexample to disprove one proposition and uses a proof by contradiction to establish the truth of two other propositions. Ultimately, it concludes that only one proposition is false, based on the properties of F' and the solutions to the equation F' = 0.