Question

Verify Lagrange’s mean value theorem for the functions on the indicated intervals. Find a point ‘c’ in the indicated interval as stated by the Lagrange’s mean value theorem:

f(x) = √(x² - 4) on [2,4]

Answer 1

Given as f(x) = √(x² - 4) on [2,4]

Since, 

f(x) exists for all the values expect (-2,2)

So, f(x) is continuous in [2,4]

f(x) = √(x² - 4)

Differentiate with respect to x

Since, also √(x² - 4) > 0

f'(x) exists for all the values of x except (2,-2)

So, f(x) is differentiable in (2,4)

Therefore, both the necessary conditions of lagrange's mean value theorem is satisfied.

So, there exist a point c ∈ (2,4)

Differentiate with respect to x 

f'(x) = x/√(x² - 4)

For the f'(c), put the value of x = c in f'(x)

f'(c) = c/√(c² - 4)

For the f'(4), put the value of x = 4 in f'(x)

For the f'(2), put the value of x = 2 in f'(x)

On squaring both sides

Thus, lagrange's mean value theorem is verified.