Application: Maximums and Minimums -- Critical Points
Let's talk about some cool stuff: finding maximums (and minimums) of a function by using its derivative.
We've already done this; earlier we used the derivative to estimate the maximum height of a missile, and also to get a sense for the longest and shortest days of the year at Northern Vermont University -- Johnson. In this section we'll crystalize our methods from these activities, and discuss a systematic process we can use to find maximums and minimums of functions with derivatives.
Before we move forward though, let's talk vocabulary:
- Technically speaking, a maximum (sometimes called a "local" maximum) of a function is an output of the function that's a little higher than all of its neighbors. For instance, when we estimated the maximum height of the missile earlier, we knew we found the maximum height because the number we found was a little higher than all of the neighboring outputs of the function.
- Sometimes when we are looking at a graph it's handy to refer to maximums as peaks instead, and we might even refer to the entire point (x- and y-coordinate) as a peak; context should make this very clear.
- A maximum is called an absolute maximum if we can somehow be sure that the maximum is as large as the function can be for any possible input. We talk about these later in this chapter.
- Similar definitions apply to the terms minimum (AKA "local" minimum), valley and absolute minimum.
If a function f has a maximum (or a minimum) at a point A, then the function f must also have stopped growing (or decreasing) at A, and consequently, f' must be equal to 0 at A.
This insight is so important, I want you to have a mental image of it. This type of thinking is what professionals who use calculus actually do. In the applet below, move A towards the peak (and valley) of f. Notice that at peaks (and valleys) of f, the slope of the tangent line, g, is 0, and so f' is equal to 0 as well.The x-coordinates of points where the derivative of a function is equal to 0 are so important that they have a special name:
f(x) are x=1 and x=3. The y-coordinates of the points where the derivative of a function is equal to 0 are called critical values. In the applet, the critical values of f(x) are y=6 and y=2, and they correspond to x=1 and x=3, respectively.
It isn't too tough to find the critical points of a function f. There's two ways to find them generally:
- Dust off your algebra skills to solve the equation formed by setting
f'(x)equal to 0. For example, iff(x)=x^3-6*x^2+9*x+2, and sof'(x)=3*x^2-12*x+9+0(be sure you can use the Monkey Rules to calculatef'on your own!), to find the critical points you'd need to solve3*x^2-12*x+9+0=0. This is hardly an impossible algebra task, but it's also not how I want to spend my weekends either. - Ask Geogebra to do it with code like we've been using:
intersect(f', y=0, LeftBound, RightBound)where you replaceLeftBoundwith the numerical value of the left extent of where you want Geogebra to hunt, andRightBoundwith the right extent of where you want Geogebra to hunt.
f(x)=x^99+sin(56x-100) or some other complex function that might rear it's ugly head in this midst of a mathematical modeling project, can you solve 99*x^98+56*cos(56x-100)=0 algebraically? I can't.
The pro of the second method is obviously that it is considerably easier, but it's not without drawbacks. For instance, what if Geogebra doesn't return any intersections?
These concerns however are for experts to discuss. For us padawans, we'll just use the second method, and if any problems come up, we'll address them as the appear.
Now, you may have already noticed a little problem: if you can't see the graph it doesn't seem clear which critical points correspond to peaks of f and which corresponds to valleys. Indeed, it's not even clear if all critical points are necessarily one or the other. Move on to the next activity to see how to deal with this kerfuffle.