Cauchy–Schwarz?

The domain of the function $f(x)=\sqrt{3x+6}+\sqrt{8-x}$ is $x=[-2,8]$ .

If we take the derivative of the function, it will become $f’(x)=-\frac{1}{2\sqrt{8-x}}+\frac{\sqrt{3}}{2\sqrt{2+x}}$ .

Then, if we equate $f’(x)$ to $0$ to find the critical points, we find that there is a relative min/max at $x=\frac{11}{2}$

To find if that is a local minimum or maximum, we can plug in the values $x=0$ and $x=6$ to $f’(x)$

At $x=0$ , the slope of $f(x)=\frac{2\sqrt{3}-1}{4\sqrt{2}} \approx 0.4356$ .

At $x=6$ , the slope of $f(x) = \frac{\sqrt{3}-2}{4\sqrt{2}} \approx -0.0473$

Since the slope goes from positive to negative, $f(x)$ is concave down . This means that $x=\frac{11}{2}$ is a relative maximum of $f(x)$

Therefore, $b=f(\frac{11}{2}) \approx \textcolor{#3D99F6}{6.324}$

To find $a$ , we must find the minimum of $f(-2)$ and $f(8)$

$f(-2) \approx 3.1623$

$f(8) \approx 5.4772$

Since $f(-2)<f(8)$ , $a=\textcolor{#3D99F6}{3.162}$

Thus, $a+b \approx 3.162+6.324 = \fbox{9.486}$

I solved it in a similar way, but used exact values. The range is $[\sqrt 10 , 2\sqrt 10].$ To avoid having to use a calculator, the problem would be better if it asked for $a * b$ instead of $a+b$ so that the answer is 20.