00:01
In this question, for the given revenue and cost functions, we are asked to find the marginal profit function.
00:09
But first we need to write down the profit function.
00:12
The profit is the difference between the revenue and the cost.
00:18
In our case, that's going to be negative q cube plus 370 q squared minus 220 minus 14 q.
00:31
Now to get the marginal profit, we need to differentiate the profit function.
00:38
The derivative of the profit function is negative 3 q squared plus 740 q minus 14.
00:48
So this is a marginal profit function.
00:53
Now we need to determine the number of items we need to sell to maximize profit.
00:59
And to maximize profit, we want the marginal profit to be equal to zero.
01:07
That means we want negative 3 q squared plus 740 q minus 14 to be equal to zero.
01:18
Let's multiply everything by negative 1.
01:20
We'll get 3 q squared minus 740 q plus 14 equals zero.
01:27
The discriminant of this quadratic is negative 740 squared minus 4 times 3 times 14.
01:38
That's going to be 547 ,432...