Lagrange Theorem Questions Solution1





In this page Lagrange theorem questions solution1 we are going to see solution of the practice questions.

(1) Verify La-grange's law of mean for the following functions:

(i) f (x) = 1 - x², [ 0 , 3]

Solution:

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

1.

f(x) is defined and continuous on the closed interval [0,3] 

2.

f(x) is differentiable on the open interval (0,3).

Then there exists at least one point c ∊ (0,3) such that 

f ' (c) = f (b) - f (a) / (b - a)

f (x) = 1 - x²

f ' (x) = 0 - 2 x

f ' (x) = - 2 x

f ' (c) = - 2 c

f (0) = 1 - 0²

        = 1

f (3) = 1 - 3²

       = 1 - 9

       = - 8

f ' (c) = f (b) - f (a) / (b - a)

         = (- 8 - 1)/(3 - 0)

         = - 9/3

    2 c = -3 

       c = -3/2  Lagrange theorem questions solution1


(ii) f (x) = 1/x , [ 1 , 2]

Solution:

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

1.

f(x) is defined and continuous on the closed interval [1,2] 

2.

f(x) is differentiable on the open interval (1,2).

Then there exists at least one point c ∊ (1,2) such that 

f ' (c) = f (b) - f (a) / (b - a)

f (x) = 1/x

f (x) = x⁻¹

f ' (x) =  (-1) x⁻²

f ' (x) = - 1/x²

f ' (c) = - 1/c²

f (1) = 1/1

        = 1

f (2) = 1/2

f ' (c) = f (b) - f (a) / (b - a)

         = (1/2) - 1/(2 - 1)

         = (-1/2)/1

         = (-1/2)

 - 1/c² = -1/2

     c² = 2

     c = ± √2

     c = √2 , - √2

- √2 ∉ (1,2)  but  √2 ∈ (1,2). So the required value of c is √2.