Page:Calculus Made Easy.pdf/101

If a curve is one that gets flatter and flatter as it goes along, the values of ${\dfrac {dy}{dx}}$ will become smaller and smaller as the flatter part is reached, as in Fig. 14.

Fig. 14.	Fig. 15.

If a curve first descends, and then goes up again, as in Fig. 15, presenting a concavity upwards, then clearly ${\dfrac {dy}{dx}}$ will first be negative, with diminishing values as the curve flattens, then will be zero at the point where the bottom of the trough of the curve is reached; and from this point onward ${\dfrac {dy}{dx}}$ will have positive values that go on increasing. In such a case $y$ is said to pass by a minimum. The minimum value of $y$ is not necessarily the smallest value of $y$, it is that value of $y$ corresponding to the bottom of the trough; for instance, in Fig. 28 (p. 101), the value of $y$ corresponding to the bottom of the trough is $1$, while $y$ takes elsewhere values which are smaller than this. The characteristic of a minimum is that $y$ must increase on either side of it.