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## AP®︎/College Calculus AB

### Course: AP®︎/College Calculus AB>Unit 5

Lesson 11: Solving optimization problems

# Optimization: profit

AP.CALC:
FUN‑4 (EU)
,
FUN‑4.B (LO)
,
FUN‑4.B.1 (EK)
,
FUN‑4.C (LO)
,
FUN‑4.C.1 (EK)
Who knows, you may end up running a shoe factory one day.  So it might not be a bad idea to know how to maximize profits. Created by Sal Khan.

## Want to join the conversation?

• What if there are more critical points in the function? The quadratic formula only gives 2 points, so how would you find the other C.P. without plugging in random numbers? •  For there to be more than two critical points, the original function would need to be x^4 or higher, which means you would have to either use the cubic formula(which is really, really long) or find some other way to turn the original expression into easier factors.
• Shouldn't Sal have checked the end behaviour of the graph first to see if there even was a maximum profit? If the graph tended towards infinity this method could have given an incorrect result right? • It is a good question, and you are mathematically right.

Now, since we deal with a factory, there are reasons to believe that past a certain point, the more you add to the production, the less it will yield. It's an economic phenomenon called "Law of diminishing returns".
So the "bunch of consultants" who came with an equation for the costs couldn't have come up with an equation where the costs are always decreasing with an increase of the production, in the first place ^^

And while this law makes intuitive sense (ask to much of somebody or something and you'll kill the goose with the golden eggs), you don't need to take it for granted. In fact, I don't think Sal didn't check the end behavior because he thought of it.

Rather, I think he knew that the general shape of a -x^3 equation tells us that as x increases, y tends towards negative infinity. With time, the general shape of these equations will pop up in your mind as you do the math.

I hope this helps as to why Sal "skipped" this step, even though you are right in pointing out that it could have been included.
(Another reason can be that Sal doesn't like to do videos of more than ten minutes and this one was already ^^)
• At , is that a local or absolute max? • Actually the global maximum depends on the interval in which it is to be checked. A plot of the functions depicts a maxima at the point and an infinite rise where x<0. Since you cannot make negative shoes, you must take the interval x>0. In this range the point is the global max. In x E R, there is no global maxima.
• Why does Sal write the first critical point to the thousandths but the second one to the ten-thousandths? • Can we use calculus to optimize a relation between workforce and profit ? like for example with 100 workers how much shoes need to be manufactured for maximum profit ? • Did you miss the 10x to find the profit? You only calculated the cost, I think... ? • It seems to me that, with this equation for profit, by giving x an arbitrarily large negative value you could get as big a profit result as you wanted. Consider: -3x^3 + 6x^2 -200x when x=-1,000,000,000. Obviously you can't make negative shoes, but I'm surprised this issue didn't show up in the example. In another equation the endless increase may be on the side of positive x-values, which means any max would not be the value at which the most profit would be made. • Okay, so before Sal solved the problem, I paused the video and took my own crack at it. Now I'm a bit confused.

While I agree with the solution derived in the video, why doesn't setting r(x) = c(x) work? That wouldn't give you profit, but the margin of profit, m(x), and setting it equal to zero would tell you at what point(s) making another shoe will incur more loss than profit. Solving it this way gives you the points x = -1, 0, and 6. The first two are out, so 6 is the answer. This can be verified by plugging 6 back into the second derivative of m(x) and getting a positive result, meaning this zero produces a minimum loss of profits (or another way of putting it is maximum gain).

So...What gives? Why doesn't this actually jive with Sal's solution?   