Mean Value Theorem





In this page mean value theorem we are going to see how to prove that between any two points of a smooth curve there is a point at which the tangent is parallel to the chord joining two points.

If f(x) be a real valued function that satisfies the following three conditions.

1.

f(x) is defined and continuous on the closed interval [a,b] 

2.

f(x) is differentiable on the open interval (a,b).

3.

f (a) = f (b)

then there exists at least one point c ∊ (a,b) such that f ' (c) = 0.

Example 1:

Using Rolle's theorem find the values of c.

f (x) =   x² - 4 x + 3 ,     1 ≤ x ≤ 3

Solution:

(i) f (x) is continuous on [1 ,3].

(ii) f(x) is differentiable (1,3).

    f(1) = 1² - 4 (1) + 3

           =  1 - 4 + 3

           =  4 - 4 

           = 0

    f(3) = 3² - 4 (3) + 3

           =  9 - 12 + 3

           =  12 - 12

           = 0

(iii) f (1) = f (3). All conditions are satisfied.

 f ' (x) = 2 x - 4 (1)

          = 2 x - 4

To find that particular point we need to set f '(x) = 0

         2 x - 4 = 0

              2 x = 4

                 x = 4/2

                 x = 2                   1 , 2 , 3


Example 2:

Using Rolle's theorem find the values of c.

f (x) =   sin² x ,     0 ≤ x ≤ Π

Solution:

(i) f (x) is continuous on [0 ,Π].

(ii) f(x) is differentiable (0,Π).

f (0) = sin² (0)

       =  0

f (Π) = sin² (Π)

       =  0

f (0) = f (Π). All conditions are satisfied.

 f ' (x) = 2 sin x cos x

          = sin 2x 

 f ' (c) = sin 2c 

To find that particular point we need to set f ' (c) = 0

          sin 2c = 0

              2c = sin⁻¹ (0)

              2c = 0 , Π , 2 Π , 3 Π , .............

               c = 0/2 , Π/2 , 2 Π/2 , 3 Π/2 , .............

               c = 0 , Π/2 ,  Π , 3 Π/2 , .............             

               c = Π/2                     0 , 2 , Π/2








Mean value Theorem to Lagrange Theorem