Page:Calculus Made Easy.pdf/134

the value of $y$ which you got must be a maximum. That’s the rule.

The reason of it ought to be quite evident. Think of any curve that has a minimum point in it (like Fig. 15), or like Fig. 34, where the point of minimum $y$ is marked $M$, and the curve is concave upwards. To the left of $M$ the slope is downward, that is, negative, and is

Fig. 34.	Fig. 35.

getting less negative. To the right of $M$ the slope has become upward, and is getting more and more upward. Clearly the change of slope as the curve passes through $M$ is such that ${\dfrac {d^{2}y}{dx^{2}}}$ is positive, for its operation, as $x$ increases toward the right, is to convert a downward slope into an upward one.

Similarly, consider any curve that has a maximum point in it (like Fig. 16 p. 82), or like Fig. 35, where the curve is convex, and the maximum point is marked $M$. In this case, as the curve passes through $M$ from left to right, its upward slope is converted