# Integral and derivative inequality [closed]

If $$f(x)$$ is derivative in the $$[0,1]$$ and $$|f'(x)|\le M$$ for all $$x\in[0,1]$$, the prove that $$\left|\frac1n\sum_{k=0}^{n-1} f(\frac kn) - \int_0^1 f(x) du\right| \le \frac M{2^n}$$

• Pointers to some basic info on writing math on Stack Exchange sites can be found here: How does one type mathematical formulas on this site? (To help you get started, I have edited your question - but I am not sure whether I read your handwriting correctly.) – Martin Sleziak Oct 15 at 13:46
• BTW I would expect the estimate to be $\frac Mn$ rather than $\frac M{2^n}$ - similarly as in this question on Mathematics: Sum approximation of a Lipschitz-continuous function. (Found using Approach0.) – Martin Sleziak Oct 15 at 13:50
• @MartinSleziak Apparently the OP has trouble even copying the homework correctly from the board: $2^n$ should obviously be $2n$. Passing next to trivial homework assignments to other people should be strongly discouraged, IMHO, so this is hardly suitable even for MSE. Voting to close. – fedja Oct 15 at 13:59