1. **Problem:** Find two numbers whose sum is 54 and whose product is a maximum. 2. **Formula and rules:** Let the two numbers be $x$ and $y$. We know: $$x + y = 54$$ We want to maximize the product: $$P = xy$$ Using the sum constraint, express $y$ in terms of $x$: $$y = 54 - x$$ So, $$P = x(54 - x) = 54x - x^2$$ 3. **Find critical points:** Differentiate $P$ with respect to $x$: $$\frac{dP}{dx} = 54 - 2x$$ Set derivative equal to zero to find critical points: $$54 - 2x = 0$$ $$2x = 54$$ $$x = 27$$ 4. **Check maximum:** Second derivative: $$\frac{d^2P}{dx^2} = -2 < 0$$ Since second derivative is negative, $x=27$ gives a maximum. 5. **Find $y$:** $$y = 54 - 27 = 27$$ 6. **Answer:** The two numbers are $27$ and $27$, and the maximum product is: $$P = 27 \times 27 = 729$$