So, I’ve had a number of people ask about whether the x-coordinate of a turning point should be included in an interval of increase or decrease or whether the intervals should always be exclusive.
Believe it or not, I’ve been tossing this question around for awhile with my good friend Brian Battistoni at Arlington High. This is a great question and it can be argued both ways. What we really need is for NYSED to just issue a clarification, so we can all be consistent. Maybe they will, maybe they won’t. I think it is a cool question because it can be argued both ways. Here’s how…
The Exclusive Case (x-coordinates of turning points not included):
This is the convention that most people will first think of because if the turning points were included, then they would have to be included in both intervals of increase and intervals of decrease, thus seemingly making the function both increasing and decreasing at the same point. This doesn’t sit well with people. On top of that, if you look at it from a calculus perspective and define the function as increasing when its slope is positive and decreasing when its slope is negative, then turning points, where the slope is either zero (smooth) or undefined (corners), should be excluded.
The Inclusive Case (x-coordinates of turning points are included):
To make and understand this argument, we need to agree on what the question actually is. Remember, the question is always phrased like “give the interval over which f(x) is increasing.” So, let’s examine the function:
Most calculus books will define an interval of increase as follows:
In other words, bigger x’s give bigger y’s. But, in the case of the quadratic above, wouldn’t the interval:
fit this definition? Remember, the question is not whether the function is increasing at x=4. It is whether x=4 is part of an interval where bigger x’s give bigger y’s. The x-coordinate of the turning point would certainly be included in this definition and is certainly included on the Advanced Placement Calculus exam. This inclusion really gets at a fundamental geometric idea regarding intervals of increase and decrease:
An interval of increase is a stretch of the function that can be considered going uphill and an interval of decrease is a stretch of the function going downhill. Aren’t the bottom and tops of hills part of these stretches?
What’s the correct definition? Well, that’s up to the bureaucrats to decide, which is weird to say. The one sample problem Albany gave us was the exclusive use. I’m amazingly inconsistent on this because of my own struggles with two equally valid interpretations.
Anyone want to weigh in?