If $a,b,c,\in \mathbf{R}$ then

 

$f\left(x\right)=\left|\begin{array}{ccc}x+a^2& ab& ac\\ ab& x+b^2& bc\\ ac& bc& x+c^2\end{array}\right|$decrease on

 

$\begin{array}{l}{f}^{\mathrm{\prime }}\left(x\right)=\left|\begin{array}{ccc}1& 0& 0\\ ab& x+b^2& bc\\ ac& bc& x+c^2\end{array}\right|+\left|\begin{array}{ccc}x+a^2& ab& ac\\ 0& 1& 0\\ ac& bc& x+c^2\end{array}\right|\\ +\left|\begin{array}{ccc}x+a^2& ab& ac\\ ab& x+b^2& bc\\ 0& 0& 1\end{array}\right|\\ =\left(x+b^2\right)\left(x+c^2\right)-b^2{c}^2+\left(x+a^2\right)\left(x+c^2\right)\\ -a^2{c}^2+\left(x+a^2\right)\left(x+b^2\right)-a^2{b}^2\\ =3x^2+2\left(a^2+b^2+c^2\right)x\\ =x\left(3x+2\left(a^2+b^2+c^2\right)\right)\end{array}$

$f$ decreases if$x < 0$ and $x>-\frac{2}{3}\left(a^2+b^2+c^2\right)$

i.e. on $\left((-2/3)\left(a^2+b^2+c^2\right),0\right)$