find two numbers whose sum is 13 and whose product is a maximum. the two numbers are (simplify your answer…

Explanation:

Step1: Set up variables

Let one number be (x), then the other number is (13 - x). The product (P=x(13 - x)=13x - x^{2}).

Step2: Find the maximum of the quadratic function

For a quadratic function (y = ax^{2}+bx + c) ((a=- 1), (b = 13), (c = 0)), the vertex of the parabola (y=ax^{2}+bx + c) has (x)-coordinate (x=-\frac{b}{2a}). Substitute (a=-1) and (b = 13) into (x =-\frac{b}{2a}), we get (x=-\frac{13}{2\times(-1)}=\frac{13}{2}). The other number is (13-\frac{13}{2}=\frac{13}{2}).

Answer:

(\frac{13}{2},\frac{13}{2})