Question

Verify Rolle’s theorem for the function f(x) = (x – a)^m(x – b)ⁿ on [a, b], m, n ∈ I⁺

Answer 1

Given f(x) = (x – a)^m(x – b)ⁿ

As we know that every polynomial function is continuous and differentiable everywhere.

So, f(x) is continuous and differentiable on the given indicated interval.

Also, f(a) = 0 = f(b).

Thus, all the conditions of Rolle’s theorem are satisfied.

Now, we have to show that there exist a point c in (a, b) such that f'(c) = 0

So,

f'(x) = m(x – a)^m–1 (x – b)ⁿ + n(x – a)^m(x – b)^n–1

f'(x) = (x – a)^m–1 (x – b)^n–1 (m(x – b) + n(x – a))

Now, f'(c) gives c = a, c = b and

⇒ (m(c – b) + n(c – a)) = 0

⇒ c = (mb  + na)/(m + n) ∈(a, b)

Hence, Rolle’s theorem is verified.