Question

The sum of two numbers is 24. Find the maximum value of their product.

Answer 1

Let x and y be the two numbers we are looking for. We are told that the sum is 24. So we have the equation

$x+y=24$

we want to calculate the maximum value of the product of both numbers, that is

$x\cdot y$

From the first equation we could replace the value of y, so we have that

$x(24\text{ -x\rparen=24x- x}^2$

which is a parabolla. Recall that the general form of a parabolla is given by the equation

$y=a(x\text{ -h\rparen}^2+k$

where (h,k) is the vertex of the parabolla. In this case, k would be the maximum or minimum value of the parabolla.

We start by factoring out the -1. So we get

$y=\text{ - \lparen x}^2\text{ -24x\rparen}$

now, we will complete the square inside the parentheses. Note that if

$(a\text{ -b\rparen}^2=a^2\text{ - 2ab + b}^2$

and we let a = x we have that

$(x\text{ -b\rparen}^2=x^2\text{ -2bx+b}^2$

If we compare this to the expression x^2 -24x, we can see that -2b=-24. So we have

$\begin{gathered} \text{ -2b= -24} \\ b=\frac{\text{ -24}}{\placeholder{⬚}\text{ -2}} \\ b=12 \end{gathered}$

So, we will add and subtract 12² so we get

$y=\text{ -\lparen x}^2\text{ -24x+12}^2\text{ -12}^2)=\text{ -\lparen x}^2\text{ -24x+12}^2)\text{ +12}^2$

which is equivalent to

$y=\text{ -\lparen x -12\rparen}^2+12^2$

by comparison, we can see that in here the value of k is 12^2. THat is

$k=12^2=144$

which is the maximum value of the product. So the correction option is the third option.