Question

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

(i) f(x) = x + 1/x, x ∈ [1, 3]

(ii) f(x) = (x² - 4)/(x - 1), x ∈ [0, 2]

Answer 1

(i) Given function

f(x) = x + 1/x, x ∈ [1, 3]

⇒ f(x) = (x² + 1)/x, 

Which is a rational function.

Since, rational function is continuous whenever its denominator will not be zero.

So, f(x) is continuous for x ≠  0.

Agin, f(x) = 1 - 1/x² = (x² - 1)/x², exists in interval (1, 3).

∴ f(x) is differentiable in (1, 3)

So, langrange's mean value theorem satisfied.

∴ A point c ∈ (1, 3) exist such that

Hence, lagrange's theorem satisfied

(ii) Given function

f(x) = (x² - 4)/(x - 1), x ∈ [0, 2]

Clearly, f(x) is continuous in interval [0, 2] and f'(x) is finite and exists. 

So,f(x) is differentiable in (0, 2). 

Hence f(x) satisfies both conditions of Langrange’s mean value theorem.

∵ c is an imaginary number.

Hence, Langrange’s mean value theorem does not satisfied.