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

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 $$\dfrac{d^2y}{dx^2}$$ 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