Quiz 11 Solutions + Clarification on Absolute vs. Local Extrema

This did not go so hot, I feel. I feel that my absence has undoubtedly influenced that, though. I do want to clarify a few things:

Absolute mins and maxes are almost entirely different from local mins and maxes. On some level, this feels intuitively wrong. After all, they are both mins and maxes. The words are so similar, that we feel that they should be the same. There are some dramatic differences in how one determines absolute extrema and local extrema, as you can see from the solutions below.

Many many of you for problem #2 substituted points into the derivative. This, unfortunately, will not yield results for what the *absolute* extrema are. Hmmm, why is that? Well, it seems like you wanted to apply the First Derivative Test determine whether they were mins or maxes. This is what you do on problem 3. But problem 3 discusses *local* extrema, not absolute extrema.

For local extrema, we don’t actually care how big the function is at that point. Local extrema, if you look closely at the definition, only cares that the f(c) we have is either smaller than a bunch of points around it (in the case of a min), or is bigger than an interval of points around c. Because we know that the first derivative of a function determines which intervals a function is purely increasing and purely decreasing, we can apply the First Derivative Test to determine whether there is this little interval where the min is smaller than everything around it. But really, we have not determined what the function’s height was at that point.

With absolute extrema, we have a theorem–the Extreme Value Theorem–which says that if I have a continuous function on a closed (and bounded) interval, then that function has an absolute minimum and it has an absolute maximum. That is, there is some tallest value of the function on that closed interval, and there is a smallest value of the function on that closed interval too.

And that’s actually pretty different from local extrema. If I wanted to find where the absolute extrema of a function on a closed interval was, I actually need to know the heights of the function! With local extrema, I just needed to know things about the derivative. Here, even if I applied the First Derivative Test, I could determine which points were local mins and local maxes, but hey, on this closed interval, there will be a biggest value and there will be a smallest value of the function. Have I found what they are yet? Not at all!

Actually, we’re making the question of absolute extrema much harder than it needs to be. I don’t need to apply the First Derivative Test in this case at all–all I need to do is plug in each critical point and each endpoint into the function, and see which value is the biggest and which is smallest. This is easier than the First Derivative Test, because for that, I would need to select two test points around each critical point and (hopefully) select them properly so that the derivative on one side has the opposite sign than at the other test point.

So, to distill it down to tl;dr form:

1. For both absolute extrema and local extrema, I will still need to differentiate the function in question, then set that function equal to zero, and solve for the critical points. That step doesn’t change from absolute to local extrema.
2. If I just care about absolute extrema, plug the critical points into the original function (*not* its derivative). I want to simply see which value is bigger or smaller, so this will work fine. Also, you need to plug in the endpoints into the function, because the critical points might not even be the biggest or smallest the function can be.
3. If I care about local extrema, I must apply the First Derivative Test. The First Derivative Test is a bit nicer, because it basically always works. With our absolute extrema strategy, we require a closed and bounded interval. The First Derivative Test doesn’t have that requirement ðŸ™‚

(You might wonder why we need the Second Derivative Test, and where that fits into this scheme. Well, like the First Derivative Test, the Second Derivative Test is about testing for local extremity. Sometimes, when a function’s second derivative is easy to compute, it is preferable to use that, rather than using the First Derivative Test and picking two test points and plugging each in.)

So that’s the rant I wanted to make ðŸ™‚

One more thing I wanted to clarify: there is a difference between the maximum and minimum’s *value*, and where that extrema is *at*. Technically speaking, in problem #2, I am asking “what is the biggest and smallest the function can be on this interval?” So that is the y value, f evaluated at that critical point (or end point). The max would be AT a particular x value. I will write the final so this will be very clear, but I did want to make you aware of this important distinction.

Also, for 1a, think about f”(c) = 0. You can still have a local minimum, but f”(c) = 0 at that c, and zero is not a positive number ðŸ™‚

Hope this helps.