Question

find two numbers whose sum is 13 and whose product is a maximum. the two numbers are (simplify your answer. use a comma to separate answers as needed.)

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:

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