(a) Show that if the profit P(x) is a maximum, then the mar- ginal revenue equals t

step 1 For this problem, we are considering profit. We're looking at p of x, which is a profit function

(a) Show that if the profit P(x) is a maximum, then...

Key Concepts

Profit Maximization

This concept involves choosing the optimal level of production that maximizes the difference between revenue and cost. In mathematical terms, this is achieved by identifying the critical points of the profit function using derivatives and confirming that these points yield a maximum profit.

Marginal Revenue and Marginal Cost

Marginal revenue is the additional income gained from selling one more unit, and marginal cost is the additional cost incurred in producing that extra unit. The condition for profit maximization often requires that these two values be equal, as producing beyond this point could decrease profit.

Cost and Revenue Functions

Cost functions express the total production costs as a function of the quantity produced, while revenue functions are derived by multiplying the price (which may depend on quantity) by the number of units sold. These functions are combined to form the profit function, which is then analyzed for optimization.

Differentiation and Critical Points

Differentiation is used to obtain the marginal functions of revenue, cost, or profit. Setting the derivative (or marginal profit) to zero yields the critical points, which are then tested, often using the second derivative test, to determine if they represent maxima or minima.

Transcript

00:01 For this problem, we are considering profit.

00:04 We're looking at p of x, which is a profit function.

00:09 And the first thing we want to do is we want to show that if i maximize profit, that the marginal revenue equals the marginal cost.

00:27 Now we want to show this is true.

00:28 So i'm going to put a question mark here.

00:30 We haven't proved this yet.

00:31 Well, when we're talking about marginal revenue, marginal cost, we're talking about the derivative.

00:37 So marginal revenue can be found by taking the derivative.

00:40 Derivative of the revenue function.

00:42 The marginal cost can be found by taking the derivative of the cost function.

00:47 So this is really what i want to show.

00:49 Is this true? well, let's go back to our profit function.

00:54 How do we define profit? well, profit can be found by starting with our revenue, how much money we bring in.

01:01 And from there, we have to subtract the money that gets paid out.

01:05 So the money i bring in, minus any money i pay out, that's what i get to keep.

01:10 That's my profit.

01:11 So if i want to maximize my profit, anytime we're finding minimums and maximums, we're going to be taking the derivative.

01:20 We'll take the derivative, set it equal to zero, and that will give us any critical points for our profit function.

01:26 It'll help us find our maximum value.

01:30 I take the derivative of my profit function.

01:33 Well, this is a subtraction.

01:34 When i take the derivative of a difference, i'm going to take the derivative of the first piece minus the derivative of the second.

01:42 Whatever those revenue and cost functions end up being.

01:46 And again, to optimize this, i'm going to set it equal to zero.

01:50 So what i have is the derivative of revenue minus the derivative of cost equals zero, or the derivative of revenue function equals the derivative of the cost function.

02:03 Yes, this is what we wanted to show.

02:06 This actually is true.

02:09 Okay, so now that we know that, let's take a look at our specific problem here.

02:14 We have been given a cost function and a demand function or a price function.

02:23 So let's write these down.

02:24 We know that our cost function is going to be 16 ,000 plus 500x minus 1 .6x squared plus 0 .004x cubed.

02:44 That's our cost function.

02:45 And we're also told that our demand function or our price function can be found here.

02:52 P of x equals 1 ,700 minus 7x.

02:58 So since i want to maximize profit, i can do, i can use what we just found.

03:05 I can find the derivative of the revenue function to find the derivative of the cost function, set them equal, and that will give me my x that will maximize my profit.

03:15 So let's do this one at a time...

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