Suppose a company's revenue function is given by r of q
and its cost function is given by c of q where q is hundreds of units sold and produced
and r of q and c of q are in dollars. Part a, we want to find the marginal profit function
which is the derivative of the profit function. And then part b,
how many items need to be sold to maximize profit and what is the maximum profit.
So before we find the marginal profit, though, we need to find the profit function
where profit is equal to revenue minus cost. So notice how here we have the revenue function
minus the cost function. Let's go ahead and simplify this. P of q is equal to negative
q to the third plus 360 q squared and then we have minus 11 q minus 480.
Now that we have the profit function, we can find the marginal profit function
which would be the derivative of the profit function or p prime of q. So the derivative
with respect to q would be negative three q squared plus 720 q minus 11. And now for part b,
how many items need to be sold to maximize the profit and what is the maximum profit? To
maximize the profit of this function, here we want to begin by determining the critical numbers which
is where the first derivative or the marginal profit function is equal to zero or undefined.
Let's begin to find the critical numbers. Notice how the derivative function is a
quadratic function which is never undefined. So to find the critical numbers, we'll set this equal
to zero and solve. Unfortunately, this is not factorable, so we'll have to use the quadratic
formula to solve this where a equals negative three, b equals 720, and c equals negative 11.
So using the quadratic formula, we would have q equals negative b or negative 720 plus or minus
the square root of b squared or 720 squared minus four times a which is negative three times c which
is negative 11 divided by two times a or two times negative three. The discriminant or the number
underneath the square root simplifies to five hundred eighteen thousand two hundred sixty-eight.
So we have two critical numbers. One is q equals negative 720 plus the square root of five
hundred eighteen thousand two hundred sixty-eight divided by negative six
and the other is q equals negative 720 minus the square root of five hundred eighteen thousand
two hundred sixty-eight divided by negative six. Let's go and get decimal approximations for these.
This first value of q is approximately 0.0153 and the second value of q is approximately 239.
So again, these are the critical numbers which should maximize or minimize the profit function
and of course we're trying to maximize the profit function. So to determine whether we
have maximum or minimum function values at these values of q, we need to use either the first or
second derivative test. Let's use the second derivative test to determine the concavity
at these values of q and let's go and do this on the next slide.
So the second derivative or p double prime of q would be equal to negative six q plus 720.
And now we'll determine the sign of the second derivative at our critical numbers to determine if
the function is concave up or down which will tell us whether we have a maximum or minimum function
value. So we'll find the sign of p double prime of 0.0153 and the sign of p double prime of 239.99.
With this small value of q here, we should be able to recognize that this function value is going to
be positive or greater than zero, which means at this critical number the function is concave up,
which means we have a low point or a minimum function value. So this is not
the value we're looking for because we're trying to maximize the profit not minimize the profit.
And now we'll find the sign of the second derivative at the second critical number.
So we'd have negative six times 239.9847
plus 720. Notice how the function value is negative and that's a good sign because that means
if the second derivative is negative, the function is concave down,
which means that this critical number we have a high point or a maximum function value.
And we are asked to round q to two decimal places. So this means the profit is maximized when q
is approximately 239. But if we go back to the original problem, remember q
is in hundreds of units. If we multiply this by 100, that means when q is approximately
this value, twenty-three thousand nine hundred ninety-eight units are sold.
So even though we found the value of q, our answer is going to be the number of units here,
which is twenty-three thousand nine hundred ninety-eight. Now they also ask for the
maximum profit, so now we need to evaluate the profit function not at this value here,
but when q is approximately 239.98. And remember our profit function was this cubic function here.
And now we'll go back to the calculator one more time.
Rounding to the nearest cent, the maximum profit would be six million
nine hundred eight thousand eight hundred eighty-eight cents.
I hope you found this helpful.