Answers:
There is a max value of 81/8 located at (x,y) = (9/8, 9)
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Explanation:
Solve the second equation for y
8x+y = 18
y = 18-8x
Plug this into the first equation
f(x,y) = x*y
g(x) = x*(18-8x)
g(x) = 18x-8x^2
This graphs out a parabola that opens downward, and has a max point at the vertex.
If you apply the derivative to this, you get g ' (x) = 18 - 16x
Set this equal to zero and solve for x
g ' (x) = 0
18 - 16x = 0
18 = 16x
16x = 18
x = 18/16
x = 9/8
Use this x value to find y
y = 18 - 8x
y = 18 - 8(9/8)
y = 18 - 9
y = 9
So the max is x*y = (9/8)*9 = 81/8
Or we could say
g(x) = 18x-8x^2
g(9/8) = 18(9/8)-8(9/8)^2
g(9/8) = 81/8
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To summarize,
There is a max value of 81/8 located at (x,y) = (9/8, 9)
When saying "max value of something", we're basically talking about the largest f(x,y) value. Which in this case is the largest x*y value based on the fact that 8x+y = 18.
A practical real world example of a problem like this would be if you wanted to max out a certain rectangular area based on constraints of how much building material you have for the fence.
