00:01
In this question, we are given the total cost and the total revenue functions, and we are asked to find the maximum revenue and the maximum profit.
00:09
Well, first we need to write down the profit function.
00:12
The profit is the revenue minus the cost.
00:20
So, in our case, that's going to be 120x minus x squared minus 950 plus 30x.
00:35
This equals to negative x squared, 120x minus 30x equals to 90x minus 950.
00:46
So that's the profit function.
00:49
Now, to maximize the revenue, we need to solve the equation r prime of x equals to 0, where our prime is the derivative or the revenue function.
01:02
So that's going to be, we need to calculate the derivative of 120x minus x squared.
01:09
And that's going to be 120 minus 2x.
01:14
That's our prime.
01:16
And we want that to be equal to 0.
01:22
After subtracting 120, we are going to get negative 2x equals to negative 120.
01:28
And after dividing by negative 2, we are going to get negative 120 over negative 2, and that's going to be 60.
01:38
So to get the maximum revenue, we must sell 60 units of the product...