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