Question 394851
**** To the OTHER tutor:  The operative assumption here is everything happens in a PERFECTLY COMPETITIVE MARKET ( which enables us to make mathematical calculations.).  So the DEMAND curve tells us that what producers manufacture actually came from the DEMAND of consumers, which THEY WILL BUY eventually.  In addition, total profit in a perfectly competitive  market reaches its maximum point where MARGINAL revenue equals MARGINAL cost.  ***





{{{c(q)= q^0.8}}} and  {{{p(q) = 5q^(-0.5)}}}
Then the profit function is {{{P(q) = qp(q) - c(q) - 2= 5q^0.5 - q^0.8 - 2}}}.
Get the derivative and set it to zero, and then solve the resulting equation: 

{{{dP/dq = 2.5q^(-0.5) - 0.8q^(-0.2) = 0}}}
==> {{{2.5q^(-0.5) = 0.8q^(-0.2)}}}
==> {{{3.125  = q^0.3}}}
==> q = 44.62, to 2 decimal places.

Use the 2nd derivative test to see if profit P is maximized at q = 44.62.
 
==> {{{d^2P/dq^2 = -1.25q^(-1.5) + 0.16q^(-1.2)}}}.  
Incidentally at q = 44.62,  {{{d^2P/dq^2 < 0}}}, and hence P is maximized.

Therefore q = 44.62 is the level of output that maximizes profit for this firm.