# Find minimum f(x)=\sin(x)+\cos(x)+\sin(2x)+1 for x \in [0,2\pi]

Find minimum $f\left(x\right)=\mathrm{sin}\left(x\right)+\mathrm{cos}\left(x\right)+\mathrm{sin}\left(2x\right)+1$ for $x\in \left[0,2\pi \right]$
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Anzante2m
Show that $f\left(t\right)=t+{t}^{2}$ for $t=\mathrm{sin}\left(x\right)+\mathrm{cos}\left(x\right)$. Now recall that $t=\sqrt{2}\mathrm{sin}\left(x+\frac{\pi }{4}\right)\in \left[-\sqrt{2},\sqrt{2}\right]$ for $x\in \left[0,2\pi \right]$. Therefore we reduced the problem to mininization of $f\left(t\right)=t+{t}^{2}$ on the interval $\left[-\sqrt{2},\sqrt{2}\right]$. The rest is clear.