(a) Show that if the profit $ P(x) $ is a maximum, then the marginal revenue equals the marginal costs.
(b) If $ C(x) = 16,000 + 500x - 1.6x^2 + 0.004x^3 $ is the cost function and $ p(x) = 1700 - 7x $ is the demand function, find the production level that will maximize profit.
a) see step for answer
b) production level of 100 maximizes profit
in part, they were going to show that if the Prophet is a maximum then the marginal revenue equals the marginal costs in perpetuity. If cfx equals 16,000 plus 500 X minus 1.6 X squared plus 0.4 X cube is the cost function. And P of x equal 1700 minus seven times sex is the demand function. Okay. Find the production level that will maximize profit. Mhm. So we start with our eight. And for that remember that the Prophet functions which depend on the production level or units produce X is equal to revenue of X minus the cost. So that that's the expression for the profit in terms of revenue and costs. So this is profit, this is revenue and the costs. And now we know that if the Prophet, this is maximum. Using the fact calculus that if the function is differentiable at a point where it has a maximum or in fact the minimum, but in this case is maximum. What we're talking about the relative get to be zero. So in this case this formula here give us that the derivative of the Prophet is equal to the derivative of the revenue minus the derivative of the cost function. But if X is a maximum or let's say it better. If the Prophet you have X is a maximum at a point X where p of X is differential, which is the case in general, because we use uh differential functions in this type of modeling, economic modelling is differentiable, then the derivative of the Prophet gallery zero. And then when that profit is maximum zero, which is a derivative of the profit will be equal to the derivative of the revenue minus the relative of the cost. That is this is always true because this is the equation we have but in the case of having a maximum profit, that is where the derivative of the profit can be zero. We will have this relationship and then derivative of the revenue equal derivative of the cost passing this term to the left. And so we have this equation and remember that the marginal revenue and the marginal ghosts are just the relative of the revenue and cost function respectively. Mm So this equation means the marginal revenue equals the marginal okay, costs. Yeah. Which is part a. Now we go for part B. So in perfect, we have the equation of the cost function. So is 16,000 plus 50 plus 500 eggs minus 1.6 X square plus zero point syria zero for X cubed. Where X is is the cost function where X is the level of production or a number of units produced. And we have to demand function P of X equal 1700. Last or minus rather find this uh seven X. This is the the man function where X is level of production. Yeah. And we want to find, okay fine the level of production that will maximize profit. And we are going to use parquet in this case because we know that the prophet is a maximum where the marginal revenue equals the marginal costs. So we're going to use for a so from bar A we have to find X. That is a level of production voyage. The margin of revenue equals the marginal marginal cost because in that case we will be at a maximum profit. In fact, if we are using the reverse of this implication here. But that's true. Also because if the marginal revenue equal the mark marks the marginal cost at a point X. It means that this derivative, this difference of derivative zero but this difference of derivatives just a derivative of the Prophet. So the derivative of the profit will be zero and the profit will be a maximum because in that case we know the profit in general has a maximum point. Mhm. So we are going to establish this equation here to find eggs and for that we need the revenue function at the cost function. The cost function we have already given And we are going to find the revenue function. And the revenue function is given us the level of production times the demand function. This is revenue function is to find like that. And so the revenue function is equal to x times 1700 minus seven X. And that is 1700 X minus seven X. Square. And now we established this equation that is we calculate first derivative after revenue function which is 1700 minus 14 X. And the derivative of the cost function is 500. Yeah 500 minus 3.2 X. Uh plus 0.0 12. That's a product of three times 0.4 X square. So the level of production we look for is dissolution to the equation to the equation derivative of revenue E conservative of cost. That is 1700 or 1700 minus 14 X. Equal 500 minus 3.2 X plus 0.0 12 X square. So we are going to arrange a little bit. This equation we get 0.0 12 X square. This term here we're going to pass all to the right and then we right the other way around. So get that. Then we have negative 3.2 X. And this past the right positive. So it's 14 minus 3.2. Just which is plus 10.8 eggs. And then we have 4 500 minus 1700 that give us um minus 1200 equals zero. This is our aggression. Mhm. And so uh to solve that we can do several things. We could for example multiply both sides by 1000. And we'll have only integer numbers but they will be some of them will be big. So we use the formula we know to solve this type of question is second degree polynomial. And so we know that the solutions of this equation because it can it can have two solutions depending on the discriminate. And this uh X is equal to negative. The coefficient of X. There is negative 10.8. More or less the square root of 10.8 square minus four times the coefficient of X square 0.0 12 times the independent term negative 1200 or 1200 or under. Sorry? Over uh two times 0.0 12. And that means that X is equal to negative 10.8 more or less squared off. And now I'm going to calculate temple eight square to 10.8 square is Uh huh 116. Yeah. Yeah point uh 64 the negative. Now I'm going to do this product of these three numbers four times uh 0.0 12 times 82,200. Give us a negative the name of plus because we have a negative here and negative here. So give us a positive sign. 5 56 57 0.6. And this is to the 0.0 24. That's the equation now. And we calculate inside the square roots we get negative 10.8 or less screwed off. Is some here is equal to uh 100 74 point any form. Mm hmm. Yeah. And we hope maybe that this girl to suspect And we are going to use a calculator to see if it is a sect. And in this case, see that is 30 13.2. So we get negative in eight 10.8 plus more or less. 13.2 over 0.0 24. So we have two possibilities. In fact we have two solutions to real solutions because the quantity inside the square root, which is called discriminates, is positive. So X one is negative 10.8 plus 13 to over 0.0 24. And that is 13.2 and 10.8. Give us 2.4 4/0 0.0 24. And we know already is going to be exact at least 100 100 times 0.0 24 is 2.4. And let's see the other solution, X two is negative. Temple eight minus 13 point 2/0 13.0.0 24. And if you do all these calculations, in fact we know it's negative. So we're going to scar the solution. But I'm going to give the number and it's 10 to 8 minus 13 2. What's your 0.0 24. This negative one South. But X represents the level of production that got to be a positive number. Since eggs must be positive, it represents level of production. Yeah, that X get B 100. It means we're going to phrase dissolution means that if the cost function is given like this. and the malfunction is this one the production level that will maximize the profits, maximize the profit is 100 units. So that's the answer to the problem here. And the key thing we use was the fact that at a maximum profit we have equality of the marginal revenue and marginal costs.