`f(x) = cos(2x), [-pi,pi]` Determine whether Rolle’s Theorem can be applied to `f` on the closed interval `[a, b]`. If Rolle’s Theorem can be applied, find all values of `c` in the open...

`f(x) = cos(2x), [-pi,pi]` Determine whether Rolle’s Theorem can be applied to `f` on the closed interval `[a, b]`. If Rolle’s Theorem can be applied, find all values of `c` in the open interval

Asked on by enotes

Textbook Question

Chapter 3, 3.2 - Problem 20 - Calculus of a Single Variable (10th Edition, Ron Larson).
See all solutions for this textbook.

1 Answer | Add Yours

shumbm's profile pic

Borys Shumyatskiy | College Teacher | (Level 3) Associate Educator

Posted on

Rolle's Theorem requires f to be defined and continuous on the given closed interval, differentiable on the open interval and values of f on ends to be equal.

Here all conditions are met (cos(-2pi)=cos(2pi)=1). Therefore there is at least one point c where f'(x)=0.

To find this point(s), find the derivative:

f'(x)=-2sin(2x). It is zero at 2x=k*pi, x=k*pi/2. There are three such points on (-pi, pi): -pi/2, 0 and pi/2.

We’ve answered 318,984 questions. We can answer yours, too.

Ask a question