Lagrange Theorem Questions Solution3





In this page Lagrange theorem questions solution3 we are going to see solution of the practice questions.

(iv) f (x) = x ^(2/3) , [-2 , 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 [-2,2] 

2.

f(x) is not differentiable on the open interval (-2,2).

Hence Lagrange theorem does not exists.


(v) f (x) = x³ - 5 x² - 3 x , [1 , 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 [1,3] 

2.

f(x) is differentiable on the open interval (1,3).

Then there exists at least one point c ∊ (1,3) such that 

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

f (x) = x³ - 5 x² - 3 x

f ' (x) = 3 x² - 5 (2 x) - 3

f ' (x) = 3 x² - 10 x - 3

f ' (c) = 3 c² - 10 c - 3

f (1) = x³ - 5 x² - 3 x

        = (1)³ - 5 (1)² - 3 (1)

        = 1 - 5 - 3

        = - 7

f (1) = - 7

f (3) = x³ - 5 x² - 3 x

       = (3)³ - 5 (3)² - 3 (3)

       = 27 - 45 - 9

       = 27 - 54

       = -27

f (3) = - 27

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

         = [-27 - (-7)]/(3 - 1)

         = [-27 + 7]/2

         = -20/2

         = -10

3 c² - 10 c - 3  = -10 

3 c² - 10 c - 3 + 10 = 0

 3 c² - 10 c + 7 = 0

 3 c² - 3 c - 7 c - 7 = 0

 3 c (c - 1) - 7 (c - 1) = 0

(c - 1) (3 c - 7) = 0

 c - 1 = 0              3 c - 7 = 0

    c = 1                  3 c = 7

                                c = 7/3   Lagrange theorem questions solution3