True or false: There exists a function f such that f(1) = −2, f(3) = 0 and f '(x) > 1 for all x ∈ R

Answer 1

We know that the slope of the secant line is (0+2)/(3-1)=1. By the intermediate value theorem, there exists point p in [1,3] such that f'(p)=1 since f is continuous and differentiable. That contradicts the f'(x)>1 everywhere statement, so such a function does not exist.