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.

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.