Step by step
The question
Question: find the maximum of f(x, y) = xy with 2x + y = 12.
1 Set up the equations
At the best point, the level curve of f just touches the curve.
So the two gradients point the same way.
The rule: ∇f = λ∇g.
λ is the Greek letter lambda.
It is just a number.
The curve is g = 12, with g(x, y) = 2x + y.
The x parts: f_x = λ g_x.
f_x is y, g_x is 2.
The y parts: f_y = λ g_y.
f_y is x, g_y is 1.
And ③: stay on the curve.
Careful!
The curve's own equation is part of the system.
Without it, we cannot find x and y.
2 Solve the equations
From ①: λ is y/2.
From ②: λ is x.
Both are λ, so they are equal.
Tidy it up: y = 2x.
Put it into ③.
Open the bracket and add the like terms.
Divide by 4: x = 3.
Then y = 2x gives y = 6.
3 Compare the values
At (3, 6), f is 18.
There is one point, so it gives the maximum: 18.
4 Check
Let's check on the xy grid.
The orange line is the curve ③: 2x + y = 12.
The blue curve is where f is 18.
It just touches the line at the point.
Another point on the line, (5, 2), gives 10.
That is smaller, as a maximum needs.
Right!
Let's look back.
First: set up the equations.
Then: solve the equations.
Then: compare the values.
Last: check.