Mean Value Theorem

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Mean Value Theorem (Lagrange Theorem). Suppose that function y=f(x) is defined and continuous on closed interval [a,b] and exists finite derivative f'(x) on interval (a,b). Then there exists point c (a<c<b) such that (f(b)-f(a))/(b-a)=f'(c).

Note, that there can be more than one such point.

Recall that (f(b)-f(a))/(b-a) is slope of line through points (a,f(a)) and (b,f(b)).

So, geometrical interpretation of the Mean Value Theorem is following: there exists such number c between a and b that tangent line at this point is parallel to the line that passes through points (a,f(a)) and (b,f(b)).

Example. If an object moves in a straight line with position function s=f(t), then the average velocity between a and b is (f(b)-f(a))/(b-a) and the velocity at t=c is f'(c). Thus, the Mean Value Theorem tells us that at some time t=c between a and b the instantaneous velocity is equal to that average velocity. For instance, if a car traveled 200 km in 2 hours, then the speedometer must have showed 100 (km)/h at least once.