Math Help - Mean Value Theorem

In the last section of Math Help online, we covered Rolles' theorem. Rolle's theorem has many uses in Mathematical analysis. In this section of Math Help, we discuss Mean Value Theorem which follows on from Rolles' theorem.

Definition of Mean Value Theorem

Mean Value Theorem is an elementary math result in mathematical analysis due to Lagrange that states that:

If a real function is continuous at every point on a closed interval [a,b] and differentiable (or has a derivative at every point) on the open interval (a,b), then there is at least a point in the open interval  c  at which the first derivative of the function   f ' (c) equals

f(b) - f(a)

b - a 

Therefore, there is a point on any arc of the graph of the function at which the tangent is parallel to the chord joining the end points of the arc.

The Generalized Mean Value Theorem or Cauchy's Mean Value Theorem

The generalized mean value theorem known as Cauchy's mean value theorem extends this to show that given two such functions, f and g, one can solve

f '(c) [ g(b) - g(a) ] = g '(c) [ f(b) - f(a) ]

for some c in [a,b].