Absolute maxima and minima Determine the location and value of the absolute extreme values of f

Absolute maxima and minima Determine the location and value...

Key Concepts

Absolute Extrema

Absolute extrema refer to the highest and lowest values that a function attains over a specified domain. In the context of a closed interval, these are the absolute maximum and absolute minimum values that the function reaches, and they are crucial for optimization problems.

Extreme Value Theorem

The Extreme Value Theorem is a fundamental result in calculus which states that if a function is continuous on a closed and bounded interval, then it must attain both a maximum and a minimum value on that interval. This theorem justifies the search for absolute extrema within a closed interval.

Critical Points

Critical points are points in the domain of a function where its derivative is either zero or does not exist. These points are candidates for local extrema, and evaluating the function at these points helps in determining potential absolute extreme values.

Endpoint Analysis

When finding absolute extrema on a closed interval, it is essential to evaluate the function at the endpoints of the interval, in addition to the critical points. This is because the absolute maximum or minimum can occur at the boundaries of the interval.

Transcript

00:01 That's done the absolute maximum and minimum values for this function, x to the two thirds times four minus x squared on the interval from negative 2 to 2.

00:09 So the first thing we need to do is find critical values.

00:12 So that's whether the derivative is equal to zero or undefined.

00:16 So it's fine the derivative.

00:18 F prime of x.

00:20 Actually, before i do that, i want to do something a little easier.

00:29 If you look at this function, the way it's written, you have to do a product rule.

00:34 There's some chain ruling involved.

00:36 So actually i'm going to do is i'm going to make this a lot simpler, and i'm going to distribute this.

00:42 So then you get 4x to the 2 thirds minus x squared times x to the 2 thirds here.

00:52 Now, if you remember your rules of exponents, you'll find that this actually simplifies to 4x to 2 thirds, minus x to the eight -thirds power.

01:07 Okay, now this is a much easier to take the derivative of.

01:11 You just have exponents, and so this is just a bunch of power rules.

01:14 We don't have to do products and chain rules, none of that.

01:17 So this is actually a much simpler derivative to take.

01:20 So now we can find it.

01:21 F prime of x is going to be two times four.

01:27 So that'll be eight thirds, x to be negative one -third, but that's the three.

01:31 The power rule minus eight thirds x to the five thirds power and now we have this eight -thirds so we're going to factor that out so we'll have eight -thirds now i'm going to do this because it's a negative exponent i'm just going to put it right into the denominator minus five or x to the five -thirds power now there are two things that we want to know here when is this equal to zero and when is it undefined anywhere? it's undefined anywhere? well, if i look at this, it's undefined right away when x equals 0.

02:23 Now, this is okay.

02:27 Some of the other examples in the past we've thought about, we don't need to worry about when it's equal to 0, but here we do, and the reason why is because when x equals 0, the original function is still defined.

02:39 If i plug in zero here for f of x, this has just become zero.

02:44 You're not dividing by zero, so there's no problem.

02:46 So actually, this function is still continuous at x when x equals zero.

02:53 So it still works.

02:54 So this actually is a critical number in this case.

02:58 So now when does it equal to zero? well, the eight thirds would, if you divide it would drop away.

03:03 Now if you want to do the algebra, you can look at this and go, you can do all the exponents, multiplying, finding common denominators, all things like that.

03:11 But if you just look at this, you think, okay, what if happens if i plug in one? okay, so then this would go to, this would be one, then you'd have one over one, so then you'd have one minus one, so one works.

03:25 So x equaling one is still a valid, is a critical number.

03:36 And then again, if you actually plug in negative one, 5 thirds would make it negative 1 here...

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