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

Integral and derivative inequality

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    $\begingroup$ 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.) $\endgroup$ – Martin Sleziak Oct 15 at 13:46
  • $\begingroup$ 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.) $\endgroup$ – Martin Sleziak Oct 15 at 13:50
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    $\begingroup$ @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. $\endgroup$ – fedja Oct 15 at 13:59