Find the range of the $\sqrt{x-3}+\sqrt{12-3x}$ We are asked to find the range of the function$$f(x)=\sqrt{x-3}+\sqrt{12-3x}$$ range
Following is solution
$$\sqrt{x-3}=\sin{t},\sqrt{4-x}=\cos{t},0\le t\le\dfrac{\pi}{2}$$it easy to find this
$$f=\sin{t}+\sqrt{3}\cos{t}=2\sin{(t+\frac{\pi}{3})}\in [1,2]$$
Is there any other quicker or smarter way to find the range of this kind of functions? (without calculus?)
Many thanks for any help!

Use Cauchy–Schwarz inequality
$$1\cdot\sqrt{x-3}+\sqrt3\sqrt{4-x}\le\sqrt{1+3}\cdot\sqrt{x-3+4-x}=2$$ 
