Here's the relevant curve I was mentioning; we only focus on the portion of the graph in between the two mentioned x values (I have included them here as blue and green "forbidden" lines).

Each time a number becomes more negative, its reciprocal becomes less negative. You can play around with actual numbers to understand this better.

This is the reason why our minimum point, which carried the most negative number possible, suddenly became the maximum point which carried the least negative number possible.