Question

If Rolle's Theorem can be applied, find all values of c in the open interval (a, b) such that f '(c) = 0. (Enter your answers as a comma-separated list. If Rolle's Theorem cannot be applied, enter NA.)

Answer 1

Since we can apply Rolle's Theorem:

`\begin{gathered} f^(\prime)(x)=-\sin (x) \\ so\colon \\ f^(\prime)(x)=0 \\ -\sin (x)=0 \end{gathered}`

Take the inverse sine of both sides:

`\begin{gathered} x=\sin ^(-1)(0) \\ x=\pi n \\ n\in\Z \end{gathered}`

Since it is for the interval:

`\lbrack\pi,3\pi\rbrack`

The solutions are:

`x=(3\pi)/(2),(5\pi)/(2)`

Answer:

`\begin{gathered} c=(3\pi)/(2),(5\pi)/(2) \\ or \\ c\approx4.71,7.85 \end{gathered}`