Question

The range of the function $f(x)=\sqrt{3-x}+\sqrt{2+x}$ is:

• A. $[2\sqrt{2},\sqrt{11}]$
• B. $[\sqrt{5},\sqrt{10}]$
• C. $[\sqrt{5},\sqrt{13}]$
• D. $[\sqrt{2},\sqrt{7}]$

Answer 1

Answer: (B)

$y^2=3-x+2+x+2\sqrt{(3-x)(2+x)}$
$=5+2\sqrt{6+x-x^2}$
$y^2=5+2\sqrt{\frac{25}{4}-{\left(x-\frac{1}{2}\right)}^2}$
$y_{\max }=\sqrt{5+5}=\sqrt{10}$
$y_{\min }=\sqrt{5}$