# Verify Lagrange’s mean value theorem for the following functions : x2-3x-1,x∈[-117,137]. - Mathematics and Statistics

Sum

Verify Lagrange’s mean value theorem for the following functions : x^2 - 3x - 1, x ∈ [(-11)/7 , 13/7].

#### Solution

The function f given as f(x) = x2 – 3x = 1 is a polynomial function. Hence, it  is continuous on [(-11)/7 , 13/7] and differentiable on ((-11)/7, 13/7).

Thus, the function f satisfies the conditions of LMVT.

∴ there exists c ∈ ((-11)/7, 13/7) such that

f'(c) = (f(13/7) - f((-11)/7))/(13/7 - ((-11)/7)   ...(1)

Now, f(x) = x2 – 3x = 1

∴ f((-11)/7) = ((-11)/7)^2 - 3((-11)/7) - 1

= (121)/(49) + (33)/(7) - 1

= (121 + 231 - 49)/(49)

= (303)/(49)
and
f(13/7) = (13/7)^2 - 3(13/7) - 1

= (169)/(49) - (39)/(7) - 1

= (169 - 273 - 49)/(49)

= (-153)/(49)

Also, f'(x) = d/dx(x^2 - 3x - 1)

= 2x – 3 x  – 0
= 2x – 3
∴ f'(c) = 2c – 3

∴ from (1), 2c – 3 = ((-153)/49 - 303/49)/(13/7 11/7)

∴ 2c – 3 = -(456)/(49) xx (7)/(24)

= (-57)/(21)

∴ 2c = (-57)/(21) + 3

= (-57 + 63)/(21)

= (6)/(21)

= (2)/(7)

∴ c = (1)/(7)∈((-11)/7 , 13/7)
Hence, Lagrange’s mean value theorem is verified.

Concept: Lagrange's Mean Value Theorem (Lmvt)
Is there an error in this question or solution?

#### APPEARS IN

Balbharati Mathematics and Statistics 2 (Arts and Science) 12th Standard HSC Maharashtra State Board
Chapter 2 Applications of Derivatives
Exercise 2.3 | Q 7.3 | Page 80