Page:Calculus Made Easy.pdf/133

Suppose, however, a case in which, like Fig. 32, the slope itself is getting greater upwards, then ${\dfrac {d\left({\dfrac {dy}{dx}}\right)}{dx}}$, that is, ${\dfrac {d^{2}y}{dx^{2}}}$ will be positive.

If the slope is becoming less as you go to the right (as in Fig. 14 p. 81), or as in Fig. 33, then, even though the curve may be going upward, since the change is such as to diminish its slope, its ${\dfrac {d^{2}y}{dx^{2}}}$ will be negative.

Fig. 33.

It is now time to initiate you into another secret–how to tell whether the result that you get by “equating to zero” is a maximum or a minimum. The trick is this: After you have differentiated (so as to get the expression which you equate to zero), you then differentiate a second time, and look whether the result of the second differentiation is positive or negative. If ${\dfrac {d^{2}y}{dx^{2}}}$ comes out positive, then you know that the value of $y$ which you got was a minimum; but if ${\dfrac {d^{2}y}{dx^{2}}}$ comes out negative, then