00:01
In this problem, we're told that the cost of producing x items is c of x equals 3x squared plus 20.
00:07
If each item is sold for 50 units, our job is to find the number of items that will maximize the profit.
00:15
So we note, first of all, that the revenue function, r of x, is price times quantity.
00:21
So in this case, it will be 50 times x.
00:25
And the profit function, we'll use pi as the problem suggests.
00:30
Pi of x for profit is the revenue minus the cost.
00:36
So we will use this idea and the given cost function to write down a function of the profit in terms of x.
00:46
This means that the profit from producing x items will be 50x minus our cost function, which is 3x squared plus 20.
00:59
So we will subtract and reorder those terms.
01:03
So we have the quadratic term first, and then the linear term, and finally the constant term.
01:12
So there is our profit expressed as a function of x.
01:16
So to maximize this, one way is to take the derivative of the profit function to find the so -called marginal profit and find the critical numbers of that function.
01:27
Function.
01:28
So the derivative of the profit function, pi prime of x, will be negative 6x plus 50.
01:36
And we want to find the so -called critical numbers, we'll abbreviate that cn.
01:41
Those are the numbers where the derivative is 0 or where the derivative does not exist...