Second Derivatives and Beyond
Stay classy, San Diego. If we have a function f that's defined on the whole real line, any extreme points must occur at critical points, as these are the only points at which we can have a peak or a valley:
Any other point can't be extreme, because the function is about to become bigger or smaller:
If we have a function defined on a closed interval, there will also be extreme points at the endpoints of that interval:
Therefore we know how to find all the interesting points (that is, points that might be extreme):
- Find all the critical points.
- If looking at a function on a closed interval, toss in the endpoints of the interval.
At this point, we have all the places where extreme points could happen. However, a critical point doesn't need to be a max or a min.
After finding all x-values where extreme points could occur, we still need to test each x-value to see whether it really is an extreme point, and if so, what type (max or min).
There are three ways to determine whether each possible extreme point you've found is a maximum, a minimum, or neither. Alas - the first way, while easiest, is usually not acceptable as an answer on exams. It can be a good way to check your work though.
- Use your graphing calculator to graph the function near the possible extreme point. Then use your eyes to see what kind of point it is.
- Use the First Derivative Test
- Use the Second Derivative Test
This math will come in handy for optimization, which is the art of classifying extreme points but with more word problems layered on top.