Step by step
The question
Question: find the minimum of f(x, y) = x² + y² with 3x + 4y = 25.
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 = 25, with g(x, y) = 3x + 4y.
The x parts: f_x = λ g_x.
f_x is 2x, g_x is 3.
The y parts: f_y = λ g_y.
f_y is 2y, g_y is 4.
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 2x/3.
From ②: λ is 2y/4.
Both are λ, so they are equal.
Tidy it up: y = 4x/3.
Put it into ③.
Open the bracket and add the like terms.
Divide by 25/3: x = 3.
Then y = 4x/3 gives y = 4.
3 Compare the values
At (3, 4), f is 25.
There is one point, so it gives the minimum: 25.
4 Check
Let's check on the xy grid.
The orange line is the curve ③: 3x + 4y = 25.
The blue curve is where f is 25.
It just touches the line at the point.
Another point on the line, (5, 5/2), gives 31.25.
That is bigger, as a minimum needs.
Right!
Let's look back.
First: set up the equations.
Then: solve the equations.
Then: compare the values.
Last: check.