# Chapter 10 - Section 10.2 - Arithmetic Sequences - Exercise Set - Page 1061: 86

The numbers are 12 and 12 and the maximum product is 144.

#### Work Step by Step

Let x be one of the two numbers. So, $24-x$ will be the other number. The product function is given by: \begin{align} & f\left( x \right)=x\left( 24-x \right) \\ & f\left( x \right)=24x-{{x}^{2}} \\ \end{align} The x-coordinate of the vertex for the equation $f\left( x \right)=24x-{{x}^{2}}$ is: \begin{align} & x=-\frac{b}{2a} \\ & x=-\frac{-24}{2\left( 1 \right)} \\ & x=-\frac{-24}{2} \\ & x=12 \\ \end{align} Therefore, \begin{align} & f\left( 12 \right)=24\left( 12 \right)-{{\left( 12 \right)}^{2}} \\ & f\left( 12 \right)=288-144 \\ & f\left( 12 \right)=144 \\ \end{align} Thus, the vertex is (12, 144). Hence, the maximum product is 144 obtained when x is 12, so other number is, \begin{align} & 24-x=24-12 \\ & =12 \end{align}

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