Second Derivatives and Beyond
Sometimes we take vacations, sometimes we take stay-cations. Second derivatives can be used to determine if the function will be traveling somewhere extreme or if it will travel somewhere more subdued. An extreme point may be either local or global.
Global means exactly that - a maximum value of a function is a global maximum if it's the biggest y-value that function ever hits. Similarly, a global minimum is the smallest y-value a function ever hits. Any maximum or minimum that isn't global is called local.
Here's a picture of some local and global extreme points:
It's possible for a function to not have a global maximum (or minimum).
The function f (x) = ex has no global maximum. Since
the function f keeps growing and growing and never hits a "biggest" value.
How can a function fail to have a global maximum? One way is if the function has the whole real line to play with, and keeps increasing all the way along the line, like these:
Another way is if the function has a vertical asymptote, in which case the function keeps increasing as x approaches (but never quite reaches) a particular value:
Another way is if the function has a hole. In this case the function tries to approach a maximum, but never quite reaches it:
The ways a function could fail to have a global minimum are similar—turn the previous pictures upside down.
Each function that fails to have a global max (or min) has one of two things going on with it. Either the function is defined on an open interval (such as the whole real line), or the function is discontinuous. If we toss out all such functions and only look at a continuous function on a closed interval, we're guaranteed to find both a global max and a global min on that interval. This is a restatement of the Extreme Value Theorem.
If we're looking at a continuous function on a closed interval, there are two places a global maximum (or minimum) can occur. It can occur in the middle of the interval, in which case it must be at a critical point:
Or it can occur at an endpoint of the interval:
If the global max occurs at a critical point, it must be the critical point with the largest function value:
This gives us a nice way to find the global max and min of a continuous function f on a closed interval [a,b].
- Find the critical points of f on [a, b].
- Find the function value at all critical points and at the endpoints x = a and x = b.
- The largest function value is the global max, and the smallest function value is the global min.
This can be quicker to do than the first and second derivative tests. We don't need to check the signs of derivatives or anything like that. We know the global max (and min) of f must happen either at a critical point or at an endpoint of the interval. So we find the value of f at critical points and endpoints, and take the biggest and smallest values.