5. Let $S=\{-3,-2,1,2,3,4\}$, and take any two numbers $a, b(a \neq b)$ from $S$. Let $g(a, b)$ be the minimum value of the function $f(x)=x^{2}-(a+b) x+a b$ with respect to the variable $x$, then the maximum value of $g(a, b)$ is $\qquad$.
5. $-\frac{1}{4}$ Detailed explanation: $g(a, b)=a b-\frac{(a+b)^{2}}{4}=-\frac{(a-b)^{2}}{4} \leqslant-\frac{1}{4}$, and when $a=1, b=2$, $-\frac{1}{4}$ is achieved.