Question

verify rolles theorem for cos(1÷x) in [-1,1]

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.


(a), function, f(x) = cos(1/x)
we have to check continuity of f(x)
we know, trigonometric functions is a continuous function. so, f(x) is continuous function.


(b) f(x) is also differentiable .
as f'(x) = -sin(1/x) × -1/x²

(c) f(x) = cos(1/x) × 1/x²
f(1) = cos(1) × 1/1² = cos1
f(-1) = cos(-1) × 1/(-1)² = cos1
hence, f(1) = f(-1)

hence, there is a point c exists in open interval (-1, 1) such that f'(c) = 0
hence varified.