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:
\(\Rightarrow\) f(x) = ex cos x on \(\big[-\frac{\pi}{2},\frac{\pi}{2}\big]\)
We know that exponential and cosine functions are continuous and differentiable on R.
Let’s find the values of the function at an extremum,
We got \(f\big(-\frac{\pi}{2}\big) = f\big(\frac{\pi}{2}\big),\) so there exist a c∈\(\big(-\frac{\pi}{2},\frac{\pi}{2}\big)\) such that f’(c) = 0.
Let’s find the derivative of f(x)
∴ Rolle’s theorem is verified.