Centuries before Newton and Leibniz, Boyer tells us that Oresme had noticed that for a semi-circle, the rate of change in the y-value per change in the x-value was is the smallest at the maximum height of the semi-circle. Similarly, Kepler, studying the best dimensions for making wine casks, noticed that as cask-volume reached its max, the change-in-volume per change-in-dimensions became less and less.
Notice that both cases are situations in which one value is the the function of something else. Anytime one value, f(x), is the function of another value, x, it allows us to think in terms of change-in-f(x) per change-in-x. That means Calculus, folks.
What both Oresme and Kepler noticed was that when the change in f(x) produced by a change in x (graphically represented as the slope of the curve) becomes zero, the maximum value of the curve is simultaneously reached.
This makes sense if you visualize what it means when the slope is equal to zero. A line with a zero slope is perfectly flat. And the smart readers among you have already learned via my Same-Same Equation that the Slope of a curve is also the Tangent. And for a semi-circle pointing up, there is only one place where the tanget to the curve is flat… at the zenith. This is the maximum value for that curve.
Note: for a semi-circle that is hanging upside-down, the slope will be flat at the nadir; in this case, finding where the derivative (the slope) equals zero will tell you where lies the minimum value of the function.
To consider more than just the semi-circle… we find that where-ever the Derivative of any continuous function equals zero, there will occur a maximum, a minimum, or what is called an “inflection point.” An inflection point occurs where the curve is changing from concave to convex or vice-versa… for that instant of transition, the slope is zero here, too. Some functions, like the sine curve, produce curves with lots of humps and hollows (peaks and troughs) so that there are many places where the derivative (slope) equals zero, meaning there are multiple maximums, minimums, or inflection points.
One last thing… and I don’t want to bring consternation upon you, but you may notice that your Calculus teachers and Calculus books will NOT ask you to find maxes and mins for curves that are convex or concave sideways (they are always opening either up or down). They don’t ask this because sideways curves scare mathematicians…
Remember what the Equation Of Same-Same tells us: that the Slope is the ratio of change-in-y per change-in-x. When the change-in-y is zero, that’s simple: flat line– you’ve reached the top or bottom of the curve (or inflection point). However, when change-in-x is zero… that’s no good… x is in the DENOMINATOR… It’s a big no-no in math to have the denom equal to zero. In such a case, the slope is said to be “undefined”– or in some cases I’ve heard professors interpret the slope as “infinity” (although that’s probably not rigorously mathematically correct). If you picture the tangent line where change-in-x equals zero (where the value of x is unchanging per value of y), it would just bea straight line up and down.