Final answer:
To maximize profit, 116 units must be produced and sold, resulting in a maximum possible profit of $13,432.
Step-by-step explanation:
To maximize profit for a product with the profit function P(x)=232x −5000−x² dollars, where x is the number of units produced and sold, one must calculate the derivative of the profit function and set it to zero to find the critical points. This represents the quantity of output that will provide the highest level of profit. Taking the derivative P'(x)=232-2x, and setting it to zero, we find that the profit-maximizing output quantity is 116 units.
Substituting x = 116 back into the original profit function, P(116)=232(116)−5000−(116)², we calculate the maximum possible profit, which gives us a profit of $13,432.
It's important to note that after finding the maximum profit point, we should also verify it provides a maximum rather than a minimum or saddle point by using the second derivative test or analyzing the nature of the curve.