You may evaluate the profit function, such that:
`p(Q) = TR(Q) - TC(Q)`
`p(Q) = 120Q - (5Q^2 + 20Q + 100)`
`p(Q) = 120Q - 5Q^2- 20Q - 100`
`p(Q) = - 5Q^2- 100Q - 100`
The first order condition for profit maximization is `(dp)/(dQ) = 0` , such that:
`(dp)/(dQ) = (d(- 5Q^2- 100Q - 100))/(dQ)`
`(dp)/(dQ) = -10Q - 100 => -10Q - 100 = 0 => -10Q = 100 => Q = -10`
The total maximized profit is obtained by substituting -10 into the equation of `p(Q) = - 5Q^2- 100Q - 100` , such that:
`p(Q) = - 5(-10)^2 - 100(-10) - 100`
`p(Q) = -500 + 1000 - 100`
`p(Q) = 400`
Hence, the total maximized profit is $400.
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