Mean Value Theorem

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

mean value theoremNote, that there can be more than one such point.

Recall that f(b)f(a)ba\frac{{{f{{\left({b}\right)}}}-{f{{\left({a}\right)}}}}}{{{b}-{a}}} is slope of line through points (a,f(a)){\left({a},{f{{\left({a}\right)}}}\right)} and (b,f(b)){\left({b},{f{{\left({b}\right)}}}\right)}.

So, a geometrical interpretation of the Mean Value Theorem is following: there exists such number cc between aa and bb that tangent line at this point is parallel to the line that passes through points (a,f(a)){\left({a},{f{{\left({a}\right)}}}\right)} and (b,f(b)){\left({b},{f{{\left({b}\right)}}}\right)}.

Example. If an object moves in a straight line with position function s=f(t)s={f{{\left({t}\right)}}}, then the average velocity between aa and bb is f(b)f(a)ba\frac{{{f{{\left({b}\right)}}}-{f{{\left({a}\right)}}}}}{{{b}-{a}}} and the velocity at t=ct=c is f(c){f{'}}{\left({c}\right)}. Thus, the Mean Value Theorem tells us that at some time t=c{t}={c} between a{a} and b{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 100kmh{100}\frac{{{k}{m}}}{{h}} at least once.