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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.)

If Rolles 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 Commaseparated List If Rolles Theorem Can class=

Sagot :

Since we can apply Rolle's Theorem:

[tex]\begin{gathered} f^{\prime}(x)=-\sin (x) \\ so\colon \\ f^{\prime}(x)=0 \\ -\sin (x)=0 \end{gathered}[/tex]

Take the inverse sine of both sides:

[tex]\begin{gathered} x=\sin ^{-1}(0) \\ x=\pi n \\ n\in\Z \end{gathered}[/tex]

Since it is for the interval:

[tex]\lbrack\pi,3\pi\rbrack[/tex]

The solutions are:

[tex]x=\frac{3\pi}{2},\frac{5\pi}{2}[/tex]

Answer:

[tex]\begin{gathered} c=\frac{3\pi}{2},\frac{5\pi}{2} \\ or \\ c\approx4.71,7.85 \end{gathered}[/tex]

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