## 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  