I have a question detailed as:
A new product is being carried at a local auto parts store. For every x thousand of the items sold, the demand function for a given price is p(x)=−5x+39 with p representing the selling price.
The cost function for the manufacturing process is c(x)=4x+30 . How many items need to be sold to produce the maximum profit for the company?
If the function for price is -5x+39, the line should be steadily decreasing from the y intercept of 39.
Likewise if the cost is 4x+30, the line should be steadily increasing from the intercept of 30.
The profit will be price-cost, but if the cost is increasing and the price is decreasing, wouldn't the 'maximum' profit be at x=1?
The solution provided is as follows:
P(x) = xp(x) – c(x)
= x(-5x + 39) – 4(x +30)
=-5x^2 + 39x -4x – 30
= 5x^2 + 35x – 30
= -5(x^2 – 7x + 6)
= -5(x-1)(x-6)
The zeros of the profit function are at x = 1 and x = 6. The vertex will be located at the midpoint of these values or $\frac{1+6}{2}=3.5$. Since x is in thousands of items, we can say that selling 3500 items will maximize the profit for the company.
I understand the factoring in each step and the conclusion, but I can't see how increasing cost and decreasing price can possibly result in a profit over time?
Is there something wrong with this question or am I interpreting it incorrectly?
I punched the first two functions and the factored result into a graphing calculator and the results seemed to not make sense for the above conclusion
Best Answer
You are right in that price is decreasing. However, the price and the revenue are two different things. That is, although the price is decreasing, you are also selling more products at this lower price, so your revenue, which is $p(x) \times x$, might be increasing. Look at the graph of $p(x) \times x$ (price * quantity = revenue) and you'll see it is initially increasing.
So technically you are right in that, if the revenue function was $-5x+39$, then the maximum profit would be at $x=0$. However, in this case $p(x) = -5x+39$ is the price function i.e. the price each item will be sold at. The revenue function would be given by price * quantity: $r(x) = xp(x)$ which is what your book solution uses.
Note how the red function which is the revenue, is initially increasing.
To find maximum profit, you have to maximize your profit function which is given by revenue-cost. This is what your provided solution is doing.