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

Sum

Verify Lagrange’s mean value theorem for the following functions : f(x) = log x on [1, e].

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Solution

The function f given as f(x) = log x is a logarithmic function which is continuous for all positive real numbers.

Hence, it is continuous on [1, e] and differentiable on (1, e).

Thus, the function f satisfies the conditions of Lagrange’s mean value theorem.

∴ there exists c ∈ (1, e) such that

f'(c) = `(f(e) - f(1))/(e - 1)`               ...(1)

Now, f(x) = log x
∴ f(1) = log 1 = 0
and
f(e) = log e = 1

Also, f'(x) = `d/dx(logx) = (1)/x`

∴ f'(c) = `(1)/c`

∴ from (1), `(1)/c`

= `(1 - 0)/(e - 1)`

= `(1)/(e - 1)`

∴ c = e – 1 ∈ (1, e)
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 7.1 | Page 80
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