Question

Verify Lagrange’s mean value theorem for the following function:

f(x) = log x on [1, e]

Answer 1

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.