Question

verify mean value theorem for f(x)= log x in interval [1 2]

Answer 1

f(x) = log x

[a,b] = [1,e]

It is a logarithmic function.

Therefore, both continuous and differentiable are in the given interval.

As it is continuous and differentiable.

Therefore, the theorem says that there is a constant  c ∈ [1,e] such that:-

f(b) - f(a)

f'(c) = b-a

Now, f'(c) = 1/c

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

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

∴ f'(c)=1/(e-1)

1/c=1/(e-1)

c=(e-1)

e ≅ 2.718

(e - 1) ≅ 1.718

1.718 ∈ [1,2.718]

(e - 1) ∈ [1,e]

Therefore, the Lagrange's mean value theorem is verified.

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