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WZ
Numerade Educator

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Problem 14 Medium Difficulty

(a) Find the intervals on which $ f $ is increasing or decreasing.
(b) Find the local maximum and minimum values of $ f $.
(c) Find the intervals of concavity and the inflection points.

$ f(x) = \cos^2 x - 2\sin x $, $ 0 \leqslant x \leqslant 2\pi $

Answer

a) Decrensing in $\left(0, \frac{\pi}{2}\right) \cup\left(\frac{3 \pi}{2}, 2 \pi\right)$ Increasing in $\left(\frac{\pi}{2}, \frac{3 \pi}{2}\right)$
b) The local maximum is $f\left(\frac{3 \pi}{2}\right)=2$
The local minimum is $f\left(\frac{\pi}{2}\right)=-2$
c) Inflection points at $x=\frac{\pi}{6}$ and $x=\frac{5 \pi}{6}$
Concave downward in $\left(0, \frac{\pi}{6}\right) \cup\left(\frac{5 \pi}{6}, 2 \pi\right)$
Concave upward in $\left(\frac{\pi}{6}, \frac{5 \pi}{6}\right)$

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Video Transcript

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