1. The problem asks for the maximum profit, which typically involves finding the maximum value of a profit function $P(x)$. 2. To find the maximum profit, we need the profit function $P(x)$, which is usually revenue minus cost: $$P(x) = R(x) - C(x)$$ where $x$ is the quantity sold or produced. 3. Once we have $P(x)$, we find its critical points by taking the derivative and setting it equal to zero: $$P'(x) = 0$$ 4. Solve for $x$ to find critical points. 5. Use the second derivative test or analyze the sign changes of $P'(x)$ to determine which critical point gives a maximum. 6. Substitute the $x$ value back into $P(x)$ to find the maximum profit. Please provide the profit function or the revenue and cost functions to proceed with calculations.