Range of radical functions.

Question

Suppose we have $f(x)=\sqrt{x-1}+\sqrt{5-x}$; how do we find range for this function? Single radicals are easy , but two of them are in this particular function.
I have the domain of the function and I can only think of differentiation to get the maximum of the function in the valid domain to find the range of $f(x)$. Is differentiation the only way, or is there something easier ?

Answer 1

By the AM-GM inequality, $$\sqrt{2+t} \sqrt{2-t} \le \frac {(2+t)+(2-t)} 2 = 2,$$

so $$(\sqrt{2+t}+\sqrt{2-t})^2 = (2+t)+(2-t)+2\sqrt{2+t}\sqrt{2-t}=4+2\sqrt{2+t}\sqrt{2-t}\le 8$$

so $$\sqrt{2+t}+\sqrt{2-t}\le\sqrt{8}.$$

Now let $x=t+3.$