Solving profit-maximizing problems under perfect competition

Solving profit-maximizing problems under perfect competition First-order conditions and second-order conditions

Situation: Perfectly competitive firm:
P = 105 => TR = PQ = 105Q
TC = 196 + 147Q - 24Q² + 2Q³
where Q = 1000s of units and TC = $1000s.

Profit = TR - TC
= 105Q - (196 + 147Q - 24Q² + 2Q³)
= -196 - 42Q + 24Q² - 2Q³

Solution:
Step1: d(Profit)/dQ = -42 + 48Q - 6Q²
Step2: FOC: -42 + 48Q - 6Q²= 0
Step3: Rearrange to solve:
- 6Q² + 48Q - 42 = 0
Solution values: Q=1,7