Page:Calculus Made Easy.pdf/102

 N.B.–For the particular value of $$x$$ that makes $$y$$ a minimum, the value of $$\dfrac{dy}{dx} = 0$$.

If a curve first ascends and then descends, the values of $$\dfrac{dy}{dx}$$ will be positive at first; then zero, as the summit is reached; then negative, as the curve slopes downwards, as in Fig. 16. In this case $$y$$ is said to pass by a maximum, but the maximum value of $$y$$ is not necessarily the greatest value of $$y$$. In Fig. 28, the maximum of $$y$$ is $$2\tfrac{1}{3}$$, but this is by no means the greatest value $$y$$ can have at some other point of the curve.

N.B.–For the particular value of $$x$$ that makes $$y$$ a maximum, the value of $$\dfrac{dy}{dx}= 0$$.

If a curve has the peculiar form of Fig. 17, the values of $$\dfrac{dy}{dx}$$ will always be positive; but there will be one particular place where the slope is least steep, where the value of $$\dfrac{dy}{dx}$$ will be a minimum; that is, less than it is at any other part of the curve.