How does a slope of zero find the best possible answer?

A slope of zero finds the best answer because at the very top of a hill, or the very bottom of a valley, the ground is level for an instant. A smooth curve that has been rising and starts to fall must pass through a flat spot, and that flat spot is where the highest point sits. In the language of calculus, if a function has a local extremum at some point and is differentiable there, then its derivative at that point must be zero.

This turns a hunt for the best into a search for flat spots. The greatest and least values a function takes are called its maximum and minimum, and finding them is the goal of mathematical optimization. Suppose you have 40 metres of fence to make a rectangular pen. A pen 19 metres long and 1 metre wide covers 19 square metres. One 15 by 5 covers 75. One 10 by 10 covers 100, and sliding away from the square in either direction makes the area shrink again. The area, plotted against the length, rises and then falls, and its slope is zero at the summit.

A zero slope is a clue, not a proof. The interior extremum theorem gives a necessary but not sufficient condition for local extrema, because some points with zero slope are not extrema at all. A level patch on the side of a hill is flat without being a summit, and a flat spot may just as well be the bottom of a valley. The second derivative, the rate at which the slope itself is changing, can often tell a maximum from a minimum. And the true global best may sit at the edge of the allowed range instead.

The method has a human story. Pierre de Fermat was one of the first mathematicians to propose a general technique for finding maxima and minima, which he called adequality. When the treatises reached René Descartes, he was doubtful of the method, but he later agreed that it was valid. Fermat's approach, which also found tangents, was equivalent to differential calculus, years before Newton and Leibniz gave calculus its famous form.