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# 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)

## Video transcript

you've opened up a shoe factory and you're trying to figure out how many thousands of pairs to shoe of shoes to produce in order to optimize your profit and so let's let X equal the thousands of pairs produced thousands of pairs pairs produced now let's think about how much money you're going to make per pair actually let me say how much revenue which is how much do you actually get to sell those shoes for so let's write a function right here revenue as a function of X well you have a whole seller who's willing to pay you \$10 per pair for as many pairs as you're willing to give him so your revenue is a function of X is going to be 10 times X and since X's and thousands of pairs produced if X is 1 that means 1,000 pairs produced times 10 which means 10,000 dollars but this will just give you 10 so this right over here is in thousands of dollars thousands of thousands of dollars so if X is 1 that means 1,000 pairs produce 10 times 1 says R is equal to 10 but that really means 10 thousands of dollars 10,000 dollars now would be a nice business if all you had was revenue and no costs but you do have costed code you have materials you have to build your factory you have to pay your employees you have to pay the electricity bill and so you hire a bunch of consultants to come up with what your cost is as a function of X what your cost is as a function of X and they come up with a function they say it is the number of the thousands of pairs you produce cubed minus 6 times the thousands of pairs you produce squared plus 15 times the thousands of pairs that you produce and once again this is going to be this is also going to be in thousands of dollars now given this this these functions of X for revenue and cost what is going to be your your what is what is profit as a function of X going to be well your profit your profit as a function of X is just going to be equal to your revenue as a function of X minus your cost as a function of X minus your cost as a function of X if you a certain amount and let's say you bring in I don't know ten thousand dollars of revenue and it costs you five thousand dollars to produce those shoes you'll have five thousand dollars in profit those numbers aren't the ones that would actually you would get from this right here I'm just giving you an example so this is what you want to optimize you want to optimize P as a function optimize P as a function of X so what is it I've just set it here in abstract terms but we know what R of X is and what C of X this is ten X this is 10 X minus all of this business so minus X to the third plus six x squared minus 15 X I just subtracted x squared you subtract a 6x squared becomes positive you subtract a 15 X it becomes negative 15 X and then we can simplify this as let's see we have negative x to the third plus 6x squared minus 15 X plus 10 X so that is minus 5 X now if we want to optimize this profit function analytically the easiest way is to think about what are the critical points of this profit function and are any of those critical points minimum points or maximum points and if one of them is a maximum point then we can say well let's produce that many that is going to be we will have optimize or we will figure out the quantity we need to produce in order to optimize our profit so to figure out critical points we essentially have to find the derivative of our function and figure out when does that derivative equal zero or when is that derivative undefined that's the definition of critical points so pre P prime of X is going to be equal to negative 3x squared plus 12x minus 5 and so this thing is going to be defined for all X so the only critical points we're going to have as when this the first derivative right over here is equal to zero so negative 3x squared plus 12x minus 5 needs to be equal to zero in order for X to be a critical point so now we just have to solve for X and so we just sort of essentially solving a quadratic equation just so that I don't have as many negatives let's multiply both sides by negative one I just like to have a clean first coefficient so if we multiply both sides by negative one we get three x squared minus 12x plus five is equal to zero and now we can use the quadratic formula to solve for X so X is going to be equal to negative B which is 12 plus or minus the square root I always need to make my radical signs wide enough the square root of B squared which is 144 minus 4 times a which is 3 times C which is 5 times 5 all of that all of that over 2a so 2 times 3 is 6 so X is equal to 12 plus or minus the square root of let's see 4 times 3 is 12 times 5 is 60 144 minus 60 is 84 all of that over 6 so X could be equal to 12 plus the square root of 84 over 6 or X could be equal to 12 minus the square root of 84 over 6 so let's figure out what these two are and I'll use a calculator I'll use I'll use the calculator for this one so I get I get let's see 12 plus the square root of 84 divided by 6 divided by 6 gives me 3.5 I'll just say 3 point 5 3 so approximately approximately 3 point I actually let me go one more let me go one more digit because I'm talking about thousand so let me say three point five two eight three point five two eight three point five to eight so this would literally be 3528 shoes because this is in thousands and let's do our pairs of shoes and then let's do the situation where we subtract and actually we could do sect we can look at our previous entry and just change this to a subtraction change that to not a negative sign a subtraction there you go and we get 0.47 - five 0.47 remember that point four seven two five approximately equal to 0.47 to five I have a horrible memory so let me review that I wrote the same thing for seven four seven two five yep all right now these are all we know about these or these are both critical points these are points at which our derivative is equal to zero but we don't know whether their minimum points their points at which the function becomes takes on a minimum value a maximum value or neither to do that I'll use the second derivative test to figure out if our function is concave upwards or concave downwards or neither at one of these points so let's look at the second derivative so P prime prime of X is going to be equal to negative 6x plus negative 6x plus 12 and so if we look if we look at let me make sure I have enough space so if we look at P prime prime P prime prime of three point five to eight so let's see if I can think this through so this is between three and four so if we take the lower value three times negative 6 is negative 18 plus 12 it's going to be less than zero and if this was four would be even more negative so this thing is going to be less than zero don't even have to use my calculator to evaluate it now what about this thing right over here 0.47 well 0.47 that's going to get us that's roughly 0.5 so negative six times 0.5 is negative three this is going to be nowhere close to being negative this is definitely going to be positive so P prime prime of zero point four seven two five is greater than zero so the fact that the second derivative is less than 0 that means that my derivative is decreasing my derivative my first derivative is decreasing when X is equal to this value which means that our graph our function is concave downwards here concave concave downwards and concave downwards means it looks it looks something like this and so and you can see when it looks something like that the slope the slope is constantly decreasing so if you have an interval where the slope is decreasing and you know the point where the slope is exactly zero which is where X is equal to three point five to eight it must be it must be a maximum it must be a maximum so we actually do take on a maximum value when X is three point five to eight on the other side we see that over here we're concave upwards concave upwards concave upwards the graph will look something like this over here and if the slope is zero where the graph looks like that we see that that is a that is a local minimum that is a local minimum and so we definitely don't want to do this we would produce for 472 and 1/2 units if we were if we were looking to minimize our profit maximize our loss so we definitely don't want to do this but let's actually think about what our profit is going to be if we produce three point five to eight thousands of shoes or 3528 shoes well to do that we just have to input it back into our original profit function right over here so let's do that so get my calculator out so my original profit function is right over there so I want to be able to see that and that so I get negative three point five to eight to the third power to the third power plus six times three point five to eight squared minus five times three point five to eight gives me and we get a drumroll now gives me a profit of give me a profit of thirteen point one two eight so let me write this down the profit when I produce 3528 shoes is approximately equal to or it is equal to if I produce exactly that many shoes it's equal to three thirteen point one two eight or actually it's approximately so I'm still rounding thirteen point one to eight so if I produce 3528 shoes and period I am going to have a profit of thirteen thousand one hundred and twenty-eight dollars remember this right over here is in thousands this right over here is thirteen point one two eight thousands of dollars in profit which is thirteen thousand one hundred and twenty eight dollars anyway we are now going to be rich shoe manufacturers