(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Let f: R->R be a function which satisfied f(0)=0 and |df/dx|≤ M. Prove that |f(x)|≤ M*|x|.

2. Relevant equations

Mean value theorem says that if f is continuous on [a,b] and differentiable on (a,b), then there is a point c such that f'(c)=[f(b)-f(a)]/(b-a).

3. The attempt at a solution

Let the derivative of f be between -M and M, and f(0)=0. For any point, p, I know that [f(p)-f(0)]/(p-0)= f(p)/p ≤ |M|.

But I don't know where to go from here...

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# Homework Help: Real Analysis Mean Value Theorem Proof

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