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Find the local maximum and minimum values of the function $$g(x)=x+2 \sin x, 0 \leq x \leq 2 \pi$$

$$g^{\prime}(x)=1+2 \cos x$$

$$1+2 \cos x=0 \rightarrow \cos x=\frac{-1}{2} \quad \longrightarrow x=\frac{2 \pi}{3} \quad, \quad \frac{4 \pi}{3}$$

The Critical numbers are $$\rightarrow 2 \pi / 3$$ and $$\frac{4 \pi}{3}$$

We have local max at $$\frac{2 \pi}{3} \longrightarrow$$

$$g(2 \pi / 3)=\frac{2 \pi}{3}+2 \sin \frac{{2\pi}}{3}=\frac{2 \pi}{3}+\sqrt{3} \approx 3.83$$

We have local min at $$\frac{4 \pi}{3} \longrightarrow$$

$$g\left(\frac{4\pi}{3}\right)=\frac{4 \pi}{3}+2 \sin \frac{4\pi}{3}=\frac{4 \pi}{3}-\sqrt{3} \approx 2.46$$

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