We have hence . Now take the integral on both sides.
I'm having some trouble with this question. I've made a start on it but I'm not sure how to finish it.
Let f be differentiable in [a,b] and let f' be continuous in [a,b]. f(a)=0. Prove the inequality in the attached image.
What I did was deal with the left side:
LHS = (b-a)(|f(b) - f(a)|) = (b-a)|f(b)|
But I don't understand why this is necessarily bigger than the RHS.