The Rolle's theorem is defined as: Suppose f is continous on $[\alpha,\beta]$ such that $f(\alpha) = f(\beta)$$f′$ exists on $(\alpha,\beta)$Then f has a local maximun or minimum at some $c \in (\alpha,\beta)$ and thus $f'(c) = 0$
And the mean value theorem is defined as: If f is continuous on $[a,b]$ and $f$' exist on $(a,b)$, then there exists some $c \in (a,b)$ such that $f'(c) = (f(b) - f(a)) /b-a$
As i see the two definitions are thesame, so the question here is that; are the two theorems thesame or their is any difference in the prove of the two theorems?
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$\begingroup$You are correct in saying that these theorems are essentially the same.
The mean value theorem is a general form of the Roll's theorem where the slope of secant is not necessarily zero.
Both theorems state that at some point the slope of tangent is the same as slope of the secant connecting the points (a , f(a) )and (b, f(b)).
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