The Mean Value Theorem, MVT, Statement and Proof

Statement of Mean Value Theorem MVT : Let a function f be

• contineous on \left[a\;,\;b\right].
• differentiable on \rbrack a,b\lbrack .

then there exist a point c\in\rbrack a,b\lbrack such that

\frac{f(b)-f(a)}{b-a}=\;f'(c)

MVT Proof: Define a new function F by

F(x)= Ax + f(x)

where A is a constant to determined such that F(a)=F(b)

The functiom Ax is contineous on \left[a\;,\;b\right] and differentiable on \rbrack a,b\lbrack . Hence F(x) is contineous and differentiable respecctively on \left[a\;,\;b\right] and \rbrack a,b\lbrack . The function F satisfies all the conditions of Rolle’s theorem therefore, there exist a point c\in\rbrack a,b\lbrack such that

F'(c)=0\;\;\;\; i.e,\;\;\; A + f'(c)=0

or f'(c)=-A

From F(a)=F(b), we have

Aa+f(a)=Ab +f(b)

or -A(b-a) =f(b)- f(a)

or -A=\frac{f(b)-f(a)}{b-c}

substituting this value of A into (1), we get

\frac{f(b)-f(a)}{b-c}=f'(c) as required.

Geometrical Interpretation of Mean Value Theorem, MVT

The mean value theorem MVT has asimple geometrical interpretation

Let the graph of f on \left[a\;,\;b\right] represented by the curve APB. Suppose the chord AB makes an angle of measure \beta with the x axis

Then

\tan\;\beta\;=\frac{BM}{AM}=\frac{f(b)-f(a)}{b-a}……………(1)

by the mean value theorem, we have

\frac{f(b)-f(a)}{b-a}=f'(c)……………(2)

where c\in\rbrack a,b\lbrack

From (1) and (2) we get \tan\;\beta\;=f'(c)

Thus at the point P (c, f(c) ), The tangent line ton the graph of y=f(x) is parallel to the chord of AB.

For physical interpretation of MVT , Suppose s= f(t) denotes the distance S that a particle has travelled at time t. If f contineous on \left[a\;,\;b\right]. and differentiable on \rbrack a,b\lbrack then by mean value theorem (mvt) there exists a point t_0\in\rbrack a,b\lbrack such that

\frac{f(b)-f(a)}{b-c}=f'(t_0)

i.e, the velocity f'(t_0) equal to the average velocity \frac{f(b)-f(a)}{b-a} from a to b.

For Example, If a car travels 400km in 5 hours then at some time the car had the speed of 80km per hour.

Also called LaGrange’s Mean Value Theorem after the name of French Mathematician J.L Lagrange(1736-1813)