1. **Problem a:** Two numbers have a sum of 29 and we want to find the two numbers that maximize their product. 2. Let the two numbers be $x$ and $y$. We know: $$x + y = 29$$ 3. Express $y$ in terms of $x$: $$y = 29 - x$$ 4. The product $P$ is: $$P = x \times y = x(29 - x) = 29x - x^2$$ 5. This is a quadratic function: $$P(x) = -x^2 + 29x$$ 6. The graph of $P(x)$ is a parabola opening downward (since the coefficient of $x^2$ is negative), so it has a maximum at its vertex. 7. The vertex $x$-coordinate is given by: $$x = -\frac{b}{2a} = -\frac{29}{2 \times (-1)} = \frac{29}{2} = 14.5$$ 8. Find $y$: $$y = 29 - 14.5 = 14.5$$ 9. Calculate the maximum product: $$P_{max} = 14.5 \times 14.5 = 210.25$$ --- 10. **Problem b:** Two numbers have a difference of 13 and we want to find the two numbers that minimize their product. 11. Let the two numbers be $x$ and $y$ such that: $$x - y = 13$$ 12. Express $y$ in terms of $x$: $$y = x - 13$$ 13. The product $P$ is: $$P = x \times y = x(x - 13) = x^2 - 13x$$ 14. This is a quadratic function: $$P(x) = x^2 - 13x$$ 15. The graph of $P(x)$ is a parabola opening upward (since the coefficient of $x^2$ is positive), so it has a minimum at its vertex. 16. The vertex $x$-coordinate is: $$x = -\frac{b}{2a} = -\frac{-13}{2 \times 1} = \frac{13}{2} = 6.5$$ 17. Find $y$: $$y = 6.5 - 13 = -6.5$$ 18. Calculate the minimum product: $$P_{min} = 6.5 \times (-6.5) = -42.25$$ **Final answers:** - a) The two numbers are 14.5 and 14.5 with maximum product 210.25. - b) The two numbers are 6.5 and -6.5 with minimum product -42.25.