Question

among all pairs of numbers whose sum is 12, find a pair whose product is as large as possible. what is the maximum product? the pair of numbers whose sum is 12 and whose product is as large as possible is (use a comma to separate answers.) the maximum product is

Explanation:

Step1: Define variables

Let one number be \(x\), then the other number is \(12 - x\). The product \(P\) is \(P=x(12 - x)=12x - x^{2}\).

Step2: Analyze the quadratic function

The function \(P(x)=-x^{2}+12x\) is a quadratic function. For a quadratic function \(y = ax^{2}+bx + c\) (\(a=- 1\), \(b = 12\), \(c = 0\)), the vertex of the parabola \(y = ax^{2}+bx + c\) has \(x\)-coordinate \(x=-\frac{b}{2a}\).
Substitute \(a=-1\) and \(b = 12\) into \(x =-\frac{b}{2a}\), we get \(x=-\frac{12}{2\times(-1)} = 6\).

Step3: Find the other number and the product

If \(x = 6\), then the other number \(12 - x=12-6 = 6\).
The product \(P=6\times6=36\).

Answer:

The pair of numbers is \(6,6\).
The maximum product is \(36\).