Question

Verify Rolle’s theorem for each of the following functions on the indicated intervals :

f(x) = sin x + cos x on [0, π/2]

Answer 1

First, let us write the conditions for the applicability of Rolle’s theorem:

For a Real valued function ‘f’:

a) The function ‘f’ needs to be continuous in the closed interval [a, b].

b) The function ‘f’ needs differentiable on the open interval (a, b).

c) f(a) = f(b)

Then there exists at least one c in the open interval (a, b) such that f’(c) = 0.

Given function is:

⇒ f(x) = sinx + cosx on \([0, \frac{\pi}{2}]\)

We know that sine and cosine functions are continuous and differentiable on R.

Let’s the value of function f at extremums:

⇒ f(0) = sin(0) + cos(0)

⇒ f(0) = 0 + 1

⇒ f(0) = 1

We have got f(0) = f(\(\frac{\pi}{2}\)). So, there exists a c∈(0, \(\frac{\pi}{2}\)) such that f’(c) = 0.

Let’s find the derivative of the function ‘f’.

∴ Rolle’s theorem is verified.