1. **State the problem:** Find two positive numbers whose sum is 300 and whose product is maximum. 2. **Set variables:** Let the two numbers be $x$ and $y$. 3. **Write the sum constraint:** $$x + y = 300$$ 4. **Express $y$ in terms of $x$:** $$y = 300 - x$$ 5. **Write the product function to maximize:** $$P = x \times y = x(300 - x) = 300x - x^2$$ 6. **Find the derivative of $P$ with respect to $x$ to find critical points:** $$\frac{dP}{dx} = 300 - 2x$$ 7. **Set derivative equal to zero to find maximum:** $$300 - 2x = 0$$ $$2x = 300$$ $$x = \cancel{\frac{2x}{2}}{\frac{300}{2}} = 150$$ 8. **Find $y$ using sum constraint:** $$y = 300 - 150 = 150$$ 9. **Check the product at $x=150, y=150$:** $$P = 150 \times 150 = 22500$$ 10. **Verify endpoints (optional):** At $x=100, y=200$, product $= 20000$; at $x=130, y=170$, product $= 22100$; at $x=125, y=175$, product $= 21875$. 11. **Conclusion:** The product is maximum when the two numbers are both 150. **Final answer:** 150, 150