## 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 