In Exercises 1-4, use the steps shown in Exercise 1 to find non- negative numbers x and y that s

step 1 Okay, for 6 .2 number 4, we're given that x plus y equals 105. We need to maximize x times y squ

In Exercises 1-4, use the steps shown in Exercise 1 to find...

Key Concepts

Optimization with Constraints

This concept refers to the process of finding the maximum or minimum of a function subject to certain restrictions. In many practical problems, such as maximizing a product or minimizing cost, the decision variables are limited by constraints that need to be satisfied, making this an essential part of optimization theory in operations research and calculus.

Substitution Method

The substitution method is used to simplify optimization problems by expressing one variable in terms of another using the given constraint. This reduces a multi-variable problem into a single-variable problem, making it more straightforward to find the optimum value using derivative-based methods.

Differentiation

Differentiation is a fundamental tool in calculus used to determine the rate at which a function is changing at any given point. In the context of optimization, differentiation helps identify critical points where the function may achieve a maximum or minimum value by setting the derivative equal to zero.

Critical Point Analysis

Critical point analysis involves finding points where the first derivative of a function equals zero or does not exist. These points are potential candidates for local maxima or minima. Further examination using the second derivative or other tests helps determine the nature of these critical points, which is central to solving optimization problems.

Transcript

00:01 Okay, for 6 .2 number 4, we're given that x plus y equals 105.

00:04 We need to maximize x times y squared.

00:09 So again, i'm following the steps that are at the beginning of this section if you want to follow along.

00:13 And i'm going to start with step three, which i believe is c in there.

00:22 If you put numbers with the letters, this is going to be step three.

00:25 So step three says that we first need to identify what we want to be minimized or maximize.

00:30 Here we want to max.

00:31 I'm going to represent that by big m for max.

00:35 And we want that to be, we want to maximize xy squared.

00:40 We also know that x plus y is equal to 105, which means y would be equal to 105 minus x.

00:52 And the purpose of doing this is because we want our m to be in terms of just x's.

00:59 So what this is really equal to is x times 105 minus x squared.

01:08 And this comes from this right here.

01:12 If you notice, i just plugged it in for y.

01:15 Okay, so this m x, y squared, is equal to, is the exact same as this right here.

01:22 Okay.

01:23 Now we have this m rewritten, and i'm going to continue to rewrite it just so it looks a little bit nicer.

01:30 So m is going to be equal to, let's see here, x times, and we're going to foil out the 105 minus x and get 11 ,025 minus 210x plus x squared.

01:52 On the next page, i'm going to distribute this x, so this is still step three.

02:00 And we are still, i'm just simplifying that function.

02:03 M is equal to 11 ,025x minus 210x squared plus x cubed.

02:18 This is the rest of step three.

02:20 So now we have our function we want to be maximized.

02:22 Step 4 is to find the domain.

02:26 And to find the domain, we're going to go back to the fact that we know that y should be equal to 105 minus x.

02:35 And y needs to be greater than or equal to zero, and that's given in the problem.

02:40 So because y is greater than or equal to zero, that implies 105 minus x, which is equal to y, also needs to be greater than or equal to zero.

02:52 So 105 is greater than or equal to x, which is the same thing as saying x is less than or equal to 105, and we know x is greater than or equal to 0.

03:05 We know x and y have to be non -negative from the question.

03:10 That's why we can say x is greater than equal to zero.

03:13 This implies that x needs to be greater than zero, but less than 105.

03:20 So this right here is what gives us our domain.

03:25 Okay, the next step that the book uses is step five, which is actually finding the critical points.

03:31 So first step 5, that's finding the critical points.

03:38 We know our m is equal to 11 ,025x minus minus 210 x squared plus x cubed.

03:55 And we want to find the critical points of this.

03:59 So to find the critical points, we first need to find the derivative.

04:03 So m prime is equal to 11 ,025 minus 420x plus 3x squared.

04:20 So the first thing you can do, this does look a little bit intimidating here, lots of big numbers...