Verify Rolle’s theorem for each of the following functions on the indicated intervals :
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:
⇒ on [ – 1,1]
We know that exponential function is continuous and differentiable over R.
Let’s find the value of function f at extremums,
⇒
⇒
⇒ f( – 1) = e0
⇒ f( – 1) = 1
⇒
⇒
⇒ f(1) = e0
⇒ f(1) = 1
We got f( – 1) = f(1) so, there exists a cϵ( – 1,1) such that f’(c) = 0.
Let’s find the derivative of the function f:
⇒
⇒
⇒
We have f’(c) = 0
⇒
⇒ 2c = 0
⇒ c = 0ϵ[ – 1,1]
∴ Rolle’s theorem is verified.