SCHWEITZER 2015 Problem 10

Let $f:\mathbb{R}\to \mathbb{R}$ be a continuously differentiable,strictly convex function.Let $H$ be a Hilbert space and $A,B$ be bounded,self adjoint linear operators on $H$ .Prove that,if $f(A)-f(B)=f'(B)(A-B)$ then $A=B$ .