Question

Find two numbers that have the maximum possible product and a sum of 7

Answer 1

Call these numbers $x,y$. Then $x+y=7$ or $y=7-x$.

We want to maximize their product,

$f(x,y)=xy\implies f(x,7-x)=F(x)=7x-x^2$

We could consider the derivative, but I think that's overkill. Instead, let's complete the square:

$7x-x^2=-\left(x^2-7x+\dfrac{49}4\right)+\dfrac{49}4=\dfrac{49}4-\left(x-\dfrac72\right)^2$

whose graph is a parabola opening downward with vertex at $\left(\dfrac72,\dfrac{49}4\right)$, so that the maximum product is $\dfrac{49}4$.

Now if $x=\dfrac72$, it follows that $y=7-\dfrac72=\dfrac72$.

Answer 2

Answer:

3.5 and 3.5

Step-by-step explanation:

The largest product comes from the most even variables.

Picture you have 7 units to make the height and width of a rectangle and your trying to get the most area possible. The most efficient rectangle is a square and all sides are even on a square so both variables should be even.