The second remark is one which (if you have any wits of your own) you will probably have already made: namely, that this much-belauded process of equating to zero entirely fails to tell you whether the ${\displaystyle x}$ that you thereby find is going to give you a maximum value of ${\displaystyle y}$ or a minimum value of ${\displaystyle y}$. Quite so. It does not of itself discriminate; it finds for you the right value of ${\displaystyle x}$ but leaves you to find out for yourselves whether the corresponding ${\displaystyle y}$ is a maximum or a minimum. Of course, if you have plotted the curve, you know already which it will be.

For instance, take the equation:

Without stopping to think what curve it corresponds to, differentiate it, and equate to zero:

whence

and, inserting this value,

will be either a maximum or else a minimum. But which? You will hereafter be told a way, depending upon a second differentiation, (see Chap. XII. p. 112). But at present it is enough if you will simply try any other value of ${\displaystyle x}$ differing a little from the one found, and see whether with this altered value the corresponding value of ${\displaystyle y}$ is less or greater than that already found.