# Practical use of max/min values, implicit differentiation.

#### Rich131

##### New member
Hi guys, this is my first post here. I will hopefully be able to help others but at the moment I have run into a problem where I don't really know where to start. I'll just get on with posting the question.

In a production process, the total production cost per day is given by the function

P = 1500x - x[SUP]2[SUB] - [/SUB][/SUP]25y/x

where x is the total number of items produced per day and y is the number of
production line workers.

On a day where each production line worker produces 25 items each, find the number
of items produced (value of x) that maximises the production cost that day.

Now what I understand of this is that I should differentiate with respect to x, but I have been trying to do implicit differentiation with no success. I used the quotient rule to solve -25y/x but my answer differs from that on WolfraamAlpha.

Any help would be greatly appreciated, specifically an explanation and direction of where to go once I have dy/dx.

#### Rich131

##### New member
Unfortunately I do not have a boss. This question comes from a management science module in a course in Management of IT/IS. I have not come across partial differentiation nor is it in the book prescribed to the module. Is it required to complete the question?

#### JeffM

##### Elite Member
Hi guys, this is my first post here. I will hopefully be able to help others but at the moment I have run into a problem where I don't really know where to start. I'll just get on with posting the question.

In a production process, the total production cost per day is given by the function

P = 1500x - x[SUP]2[SUB] - [/SUB][/SUP]25y/x

where x is the total number of items produced per day and y is the number of
production line workers.

On a day where each production line worker produces 25 items each, find the number
of items produced (value of x) that maximises the production cost that day.

Now what I understand of this is that I should differentiate with respect to x, but I have been trying to do implicit differentiation with no success. I used the quotient rule to solve -25y/x but my answer differs from that on WolfraamAlpha.

Any help would be greatly appreciated, specifically an explanation and direction of where to go once I have dy/dx.
Have you given the problem exactly? Does it say anything about implicit differentiation?

It is given that $$\displaystyle \dfrac{x}{y} = 25.$$

If your cost function is $$\displaystyle p = 1500x - x^2 - 25\left(\dfrac{y}{x}\right),$$

$$\displaystyle then\ p =1500x - x^2 - 25\left(\dfrac{1}{\frac{x}{y}}\right) = 1500x - x^2 - 25\left(\dfrac{1}{25}\right) = 1500x - x^2 - 1.$$

Now differentiate p with respect to x.

I deleted my first post. No, you do not need to do partial differentiation because x / y is given.

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#### Rich131

##### New member
Thank you!

I don't know how I missed x/y being given to me in the question. I assumed (foolishly) that I would have to implicitly differentiate as the questions were getting progressively more difficult and the one before it was implicit.

So all I have to do now is differentiate

1500x - x[SUP]2[/SUP] - 1

and let it = 0 to get x?

Also, how do you get the math style text? Thanks again Jeff!

#### JeffM

##### Elite Member
I don't know how I missed x/y being given to me in the question. I assumed (foolishly) that I would have to implicitly differentiate as the questions were getting progressively more difficult and the one before it was implicit.

So all I have to do now is differentiate

1500x - x[SUP]2[/SUP] - 1

and let it = 0 to get x? Yes

Also, how do you get the math style text? Thanks again Jeff!
Well I missed it too, but before I could delete my post to correct myself, you had already responded.

If you click on my original post, you will see the coding required to get some math style text. Unless you are going to ask a lot of questions, it probably is not worth your time to learn LaTeX coding though the tutors will be grateful if you do.

#### Rich131

##### New member
$\frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Yeah that took longer than I expected, main issue was I wrapped it in CODE tags.. I suppose it's relatively useful to know right?