5. Let $S=\{-3,-2,1,2,3,4\}$, and take any two numbers $a, b$ from $S$, where $a \neq b$. Then the maximum value of the minimum value of the function $f(x)=x^{2}-(a+b) x + a b$ is $\qquad$ .
5. $-\frac{1}{4}$.

It is easy to see that the minimum value of $f(x)$ is
$$
\varphi(a, b)=-\frac{1}{4}(a-b)^{2} \text {. }
$$

Since $a, b \in S$, and $a \neq b$, the minimum value of $|a-b|$ is 1. Therefore, the maximum value of $\varphi(a, b)$ is $-\frac{1}{4}$.