To find the range of $\sqrt{x-1} + \sqrt{5-x}$ To find the range of $\sqrt{x-1} + \sqrt{5-x}$.
I do not know how to start?
Thanks

Clearly $1\le x\le5$
As $\dfrac{1+5}2=3, 1-3\le x-3\le5-3$
WLOG $x-3=2\cos2y,$  where $0\le2y\le\pi$
$\sqrt{x-1}=2\cos y$  and $\sqrt{5-x}=2\sin y$
Now $\sin y+\cos y=\sqrt2\sin\left(y+\dfrac\pi4\right)$ and $0\le y\le\dfrac\pi2$
$\implies\dfrac1{\sqrt2}\le\sin\left(y+\dfrac\pi4\right)\le1$
