Four Real Roots

The curve of $f(x)=|x^2-3|+k-11$ is symmetrical along the y-axis. We note the following:

• It has two minima at $x=\pm \sqrt{3}\quad \Rightarrow f(\pm \sqrt{3} )= k - 11$
• It has a local maximum at $x=0\quad \Rightarrow f(0) = k - 8$
• As $k$ increases, $f(x)$ is shifted upward.

We note that when $k=8$ , then $f(0) = 8-8=0$ ; that is the local maximum is tangent to the x-axis and $f(x)$ has three roots. Therefore $a=8$ .

When $k=11$ , the minima $f(\pm \sqrt{3} )= 11-11=0$ are tangent to the x-axis and $f(x)$ has only two roots. Therefore $b=11$

$\Rightarrow a+b = \boxed{19}$

The following figure shows $f(x)$ for $k=7$ to $k=12$ .