Mathematics
Mathematics
Problem
A firm’s profit function is differentiable for all production levels and its derivative is given by . Which of the following best justifies that producing 40 units yields a maximum profit for the firm? a) Because and , indicating the profit curve is concave up at . b) Because and , indicating the profit curve is concave down at . c) Because and for all , so profit decreases once production exceeds 40 units. d) Because is zero only at , making 40 units the absolute maximum by uniqueness of the critical point.Intuitive explanation
The key is recognizing that maximum profit occurs where marginal profit equals zero — that's where . Once you find that critical point, you need to verify it's actually a maximum by checking that the derivative changes from positive to negative as increases through that point. This tells you profit is rising before that production level and falling after, confirming a peak.
Intuitive explanation
The key is recognizing that maximum profit occurs where marginal profit equals zero — that's where . Once you find that critical point, you need to verify it's actually a maximum by checking that the derivative changes from positive to negative as increases through that point. This tells you profit is rising before that production level and falling after, confirming a peak.