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?!