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.