# Math Help - Demand and Total Cost Curves

1. ## Demand and Total Cost Curves

I am given that:
Q=100-P
TC = 1/3Q³-7Q²+111Q+50

Find output and price that maximise profit and then calculate profit. Ensure you have found a maximum and checked for boundary maxima.

So far I have done this:

Rearranged Q so that
P= -Q+100

Q*P = TR = -Q+100 (Q) = -Q²+100Q

Profit = TR-TC= -Q²+100Q-1/3Q³-7Q²+111Q+50
=1/3Q³-6Q²-11Q+50

First Derivative = -Q²-12Q-11
(-Q-11)(Q+1)
Therefore Q=11 or Q=-1

Then Used Second Deriv to work out Min/Max:
=-2Q-12
When Q=11, Maximum
When Q=-1, Maximum

Now I am unsure If I have made a mistake up to this point however I sort of come stuck here. Does anyone know if I have made any mistakes and what the next step would be?

My thought is that you cant have a negative quantity and therefore Q must = 11. When Q = 11, P=89

and then i was just going to plug the figures into the Profit formula and see what i get?!

2. Assuming your algebra is correct, You are right to discount Q=-11.

a maximum can occur at either:

Q=0
Q=100 (corresponds to P=0)
Or somewhere in between.

Your differenciation has given you the maximum profit in the "somewhere in between" range. "Check for boundary maxima" means check that this gives higher profits than Q=0 or Q=100.