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How to prove the Mean Value Theorem using Rolle's Theorem? I am getting the impression that it is possible by adding a linear function to a function where Rolle's theorem applies to prove the MVT. However, I can't quite turn this idea into a rigorous mathematical argument.

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    $\begingroup$ Use the function defined by $\phi(x)=f(x)-f(a)-{f(b)-f(a)\over b-a}(x-a)$ (this gives the difference of $f$ and the function whose graph is the line segment joining the points $\bigl(a,f(a)\bigr)$ and $\bigl(b, f(b)\bigr)$. $\endgroup$ – David Mitra Dec 15 '12 at 1:33
  • $\begingroup$ @David I think I understand. If I subtract the line from the function, then the point c between a and b such that $f'(c) = (f(b) - f(a))/(b - a)$ becomes $f'(c) = 0$. However, how can I now mathematically use that function to formally prove the MVT? $\endgroup$ – hesson Dec 15 '12 at 1:45
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    $\begingroup$ My analysis professor's one-sentence proof was "tilt your head". $\endgroup$ – MJD Dec 15 '12 at 3:00
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For $f$ continuous on $[a,b]$ and differentiable on $(a,b)$, the standard proofs I've seen use the function that gives the difference of $f$ and the function whose graph is the line segment joining the points $\bigl(a,f(a)\bigr)$ and $(b,f(b)\bigr)$; $$ \phi(x)=f(x)-f(a)-{f(b)-f(a)\over b-a}(x-a). $$

From the continuity and differentiablity of $f$ (and standard theorems such as the difference of continuous functions is continuous) it follows that $\phi$ is continuous on $[a,b]$ and differentiable on $(a,b)$. Since $\phi(a)=\phi(b)=0$, Rolle's theorem applies to $\phi$ on $[a,b]$. Writing down the result obtained from Rolle's Theorem gives all that is desired.

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