# How do you determine whether Rolle's theorem can be applied to #f(x) = (x^2 - 1) / x# on the closed interval [-1,1]?

Rolles theorem states: "Let f be continuous on a closed interval [a, b] and differentiable on the open interval (a, b). If f(a) = f(b), then there is at least one point c in (a, b) where f '(c) = 0."

The first criteria is NOT met. Specifically, f is NOT continuous on the interval [-1, 1] at the point x = 0.

At x = 0, the function is undefined.

Hope that helps.

By signing up, you agree to our Terms of Service and Privacy Policy

Rolle's theorem can be applied to a function ( f(x) ) on a closed interval ([a, b]) if the following conditions are met:

- ( f(x) ) is continuous on ([a, b])
- ( f(x) ) is differentiable on ((a, b))
- ( f(a) = f(b) )

For the function ( f(x) = \frac{x^2 - 1}{x} ) on the closed interval ([-1, 1]), we need to check these conditions:

- ( f(x) ) is continuous on ([-1, 1]) since it is a rational function and is continuous everywhere except at ( x = 0 ), which is not in the interval ([-1, 1]).
- ( f(x) ) is differentiable on ((-1, 1)) since it is a rational function and is differentiable everywhere except at ( x = 0 ), which is not in the interval ((-1, 1)).
- ( f(-1) = \frac{(-1)^2 - 1}{-1} = \frac{1 - 1}{-1} = 0 ) and ( f(1) = \frac{1^2 - 1}{1} = \frac{1 - 1}{1} = 0 ).

Since all three conditions are met, Rolle's theorem can be applied to the function ( f(x) = \frac{x^2 - 1}{x} ) on the closed interval ([-1, 1]).

By signing up, you agree to our Terms of Service and Privacy Policy

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

- How do use the first derivative test to determine the local extrema #36x^2 +24x^2#?
- How do you verify that #y = x^3 + x - 1# over [0,2] satisfies the hypotheses of the Mean Value Theorem?
- How do you find the coordinates of relative extrema #f(x)=x^3-4x^2+x+6#?
- Using the principle of the mean-value theorem on the indicated interval, how do you find all numbers c that satisfy the conclusion of the theorem #f(x)=x-cosx#; [-pi/2, pi/2]?
- What are extrema and saddle points of #f(x,y)=(x+y+1)^2/(x^2+y^2+1)#?

- 98% accuracy study help
- Covers math, physics, chemistry, biology, and more
- Step-by-step, in-depth guides
- Readily available 24/7