Find minimum $f\left(x\right)=\mathrm{sin}\left(x\right)+\mathrm{cos}\left(x\right)+\mathrm{sin}\left(2x\right)+1$ for $x\in [0,2\pi ]$

jamessinatraaa
2022-01-14
Answered

Find minimum $f\left(x\right)=\mathrm{sin}\left(x\right)+\mathrm{cos}\left(x\right)+\mathrm{sin}\left(2x\right)+1$ for $x\in [0,2\pi ]$

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Anzante2m

Answered 2022-01-15
Author has **34** answers

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

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