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Found problems: 1

2006 Victor Vâlcovici, 2
Tags: derivative , lipschitz continuous , riemann sum , calculus , function , real analysis
Let be a differentiable function $ f:[0,1]\longrightarrow\mathbb{R} $ whose derivative has a positive Lipschitz constant $ L. $ Show that [b]a)[/b] $ x,y\in [0,1]\implies | f(x)-f(y)-f'(y)(x-y) |\le L\cdot (x-y)^2 $ [b]b)[/b] $ \lim_{n\to\infty } \left( n\int_0^1 f(x)dx-\sum_{i=1}^nf\left( \frac{2i-1}{2n} \right) \right) =0. $