Answer :
Answer:
8 and 8; 64
Step-by-step explanation:
1. x + y = 16
2. y = 16 - x
3. Product = xy = x (16 - x) = -[tex]x^{2}[/tex] + 16x = f(x)
4. x-coordinate of the vertex = [tex]\frac{-16}{2(-1)} =8[/tex]
5. y = 16 - 8 = 8
6. xy = 8(8)= 64
The pair of numbers that maximizes the product is x = 2 and y = 2.
How to find the maximum product?
Let's define two numbers x and y, such that the sum is 4, so:
x + y = 4.
The product of these two numbers is given by:
P = x*y
Now, using the first equation we can isolate one of the variables, I will isolate x so we get:
x = 4 - y
Now we can replace that in the product to get:
P = (4 - y)*y = 4y - y^2
Notice that this is a parabola of negative leading coefficient, so the maximum is at the vertex. Using the general formula for the vertex of a parabola we get:
y = -4/(2*-1) = 2
Now that we know the value of y, we can use:
x = 4 - y = 4 - 2 = 2
So we know the values of the two numbers, y = 2 and x = 2 give the maximum product.
If you want to learn more about maximization, you can read:
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