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