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.