First Derivative TestNote the 3 critical numbers (the dotted lines). At each of them, the derivative graph has a root. At the first, the derivative graph changes from positive to negative, and the original function has a local maximum. At the second, the derivative graph does not change sign, and the original function has neither a maximum nor a minumum. At the third, the derivative changes from negative to positive and the function has a local minumum. This indirect method of analyzing a critical number is seldom needed in practice, now that we have good graphing software, but for some rather touchy functions, it is still a useful tool. Also, it is important because future theorems will build on it. Dave Matthews, Created with GeoGebra |