1. **State the problem:** Find two positive numbers $x$ and $y$ such that $x + 2y = 100$ and the product $P = xy$ is maximized. 2. **Express one variable in terms of the other:** From the constraint, $x + 2y = 100$, solve for $x$: $$x = 100 - 2y$$ 3. **Write the product function:** $$P = xy = (100 - 2y) y = 100y - 2y^2$$ 4. **Maximize the product:** To find the maximum, take the derivative of $P$ with respect to $y$ and set it to zero: $$\frac{dP}{dy} = 100 - 4y = 0$$ 5. **Solve for $y$:** $$100 - 4y = 0 \implies 4y = 100 \implies y = \frac{100}{4} = 25$$ 6. **Find $x$ using $y=25$:** $$x = 100 - 2(25) = 100 - 50 = 50$$ 7. **Check the product:** $$P = xy = 50 \times 25 = 1250$$ 8. **Verify maximum with second derivative:** $$\frac{d^2P}{dy^2} = -4 < 0$$ which confirms a maximum. **Final answer:** The two positive numbers are $x = 50$ and $y = 25$. Note: The answer $2 \frac{1}{4}$ given does not match the problem's solution for maximum product under the given constraint.