Lagrange Theorem Questions





In this page Lagrange theorem questions we are going to see some practice questions. For each question you can find a solution with detailed explanation.

Example :

Using Lagrange theorem find the values of c.

f (x) =   2x³ + x² - x - 1,      0 ≤ x ≤ 2

Solution:

(i) f (x) is continuous on [0 ,2].

(ii) f(x) is differentiable (0,2).

f ' (x) = 2(3x²) + 2 x - 1 - 0

         = 6 x² + 2 x - 1 

f (x) =   2x³ + x² - x - 1

f (0) = 2 (0)³ + (0)² - 0 - 1

f (0) = -1

f (2) = 2 (2)³ + (2)² - 2 - 1

       = 2 (8) + 4 - 2 - 1

       = 16 + 4 - 3

       = 20 - 3

f (0) = 17

Here a = 0 and b = 2

 f ' (c) = [f (b) - f (a)] / (b - a)

6 x² + 2 x - 1  = [17 - (-1)]/(0 + 2) 

6 x² + 2 x - 1  = (17 +1)/2

6 x² + 2 x - 1  = 18/2

6 x² + 2 x - 1  = 9

6 x² + 2 x - 1 - 9 = 0

6 x² + 2 x - 10 = 0

3 x² +  x - 5 = 0

We cannot solve this equation by using factorization method. So we are trying to solve this equation by using quadratic formula.

 a = 3, b = 1, c = -5     

    x = - 1 ± √ [(1)² - 4 (3) ( -5 )]/2 (3)

    x = - 1 ± √ [1 + 64]/6

    x = [- 1 ± √(65)]/6

    x = (- 1 ± 8.06)/6

    x = (- 1 + 8.06)/6 ,  x = (- 1 - 8.06)/6

    x = 7.06/6 ,  x = - 9.06/6

    x = 1.17  ,  x = - 1.51

now let us see some practice questions of this topic.


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

(i) f (x) = 1 - x², [ 0 , 3]    Solution

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

(iii) f (x) = 2 x³ + x² - x - 1 , [0 , 2]    Solution

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

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

(2) If f (1) = 10 and f ' (X)  ≥ 2 for 1 ≤ x ≤ 4 how small can f (4) possibly be?      Solution

(3) At 2.00 p.m car's speedometer reads 30 miles/hr., at 2. 10 pm it reads 50 miles/hr. Show that sometime between 2.00 and 2.10 the acceleration is exactly 120 miles/hr²     Solution      Lagrange theorem questions








Lagrange Theorem to Mean Value Theorem