Can you do that without a graph?

Geometry Level 5

Without using graph technique, find the minimum value of $f(x)=\sin x+\sqrt{3} \cos x - \sin 2x$ .

Let $M$ denote the minimum value, submit $\lfloor 1000M \rfloor$ .

1 solution

Ryan S
Jun 7, 2020

$f(x)=2\sin\left(x+\frac\pi3\right)-\sin(2x)$

If $f$ is minimum at $x$ , $f^\prime(x)=2\cos\left(x+\frac\pi3\right)-2\cos(2x)=0$

$\cos\left(x+\frac\pi3\right)=\cos(2x)$

$\pm2x=x+\frac\pi3+2\pi n$ for $n\in\mathbb Z$

$x=\frac{\frac\pi3+2\pi n}{\pm2-1}$

Then there are only four answers to explore