# Verify Lagrange’s mean value theorem for the following functions : f(x) = x-1x-3 on [4, 5]. - Mathematics and Statistics

Sum

Verify Lagrange’s mean value theorem for the following functions : f(x) = (x - 1)/(x - 3) on [4, 5].

#### Solution

The function f given as f(x) = (x - 1)/(x - 3) is a rational function which is continuous except at x = 3.
But 3 notin [4, 5]
Hence, it is continuous on [4, 5] and differentiable on (4, 5).
Thus, the function f satisfies the conditions of Lagrange’s mean value theorem.
∴ there exists c ∈ (4, 5) such that

f'(c) = (f(5) - f(4))/(5 - 4)                   ...(1)

Now, f(x) = (x - 1)/(x - 3)

∴ f(4) = (4 - 1)/(4 - 3) = (3)/(1) = 3

and f(5) = (5 - 1)/(5 - 3) = (4)/2) = 2

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

= ((x - 3).d/dx(x - 1) - (x - 1).d/dx(x - 3))/(x - 3)^2

= ((x - 3) xx (1 - 0) - (x - 1) xx (1 - 0))/(x - 3)^2

= (x - 3 - x + 1)/(x - 3)^2

= (-2)/(x - 3)^2

∴ f'(c) = (-2)/(c - 3)^2

∴ from (1), (-2)/(c - 3)^2

= (2 - 3)/(1)
= – 1
∴ (c – 3)2 = 2
∴ c – 3 = ±sqrt(2)
∴ c = 3 ±sqrt(2)
But (3 - sqrt(2)) notin (4, 5)
∴ c ≠ 3 - sqrt(2)
∴ c = 3 + sqrt(2) ∈ (4, 5)
Hence, Lagrange’s mean value theorem is verified.

Concept: Lagrange's Mean Value Theorem (Lmvt)
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#### 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 7.5 | Page 80