SCHWEITZER 1975 Problem 6

Let $f$ be a differentiable real function and let $M$ be a positive real number. Prove that if $$|f(x+t)-2f(x)+f(x-t)| \leq Mt^2 \textrm{for all}\ x\ \textrm{and}\ t\ ,$$ then $$|f'(x+t)-f'(x)| \leq M|t|.$$

J. Szabados