Question

verify lagrange mean value theorem for the function f of X equal to log x on [1,e]

Answer 1

Step-by-step explanation:

f(x) = log x

[a,b] = [1,e]

Since it is a logarithmic function, it is both continuous and differentiable in the given interval.

If it is continuous and differentiable, then the theorem states that there exists a constant  c ∈ [1,e] such that

$f'(c) = \frac{f(b) - f(a)}{b - a}$

Here, f'(c) = $\frac{1}{c}$

f(b) = f(e) = log e = 1

f(a) = f(1) = log 1 = 0

∴ $f'(c) = \frac{1}{e - 1}$

⇒ $\frac{1}{c} = \frac{1}{e - 1}$

⇒ c = (e - 1)

e ≅ 2.718

(e - 1) ≅ 1.718

1.718 ∈ [1,2.718] ⇒ (e - 1) ∈ [1,e]

Hence, the Lagrange's mean value theorem is verified