I'm trying to explain how to use Lagrangian multipliers, through an example.
I start in this way:
Understanding how Lagrangian multipliers work can be done through a simple example. Consider the function: $$ f(x,y) = -2x^2-3y^2 + 3$$ The constraint equation will be defined as the following plane: $$ c(x,y) = x + y = 1 $$ Using a Lagrangian multiplier, we can find the highest point which lies both on the function and the plane.
The intersection line between the plane and the function $f(x,y)$ is graphed in the following plot in red:
My question is:
In this graph, the gradient vector (arrows in blue) for the function $f(x,y)$ shows the direction of steepest ascent of the function. But what does the gradient vector of the intersection line (arrows in orange) point towards? Highest gradient of $f(x,y)$ (and not increase in height) ? - and in a way that its perpendicular to the intersection line?
Also: is the fact that the magnitude of the gradient vectors I drew are slowly decreasing as it approaches the top of the paraboloid correct?
Thank you for any help. I am a highschool student writing a paper about calc. of variations, so my knowledge is very very basic (so please explain in detail if you have time).
The Vectors
The gradient vector is the direction in which the function increases the most.
The blue gradient vectors, indicates the direction to move, if you want to increase the value of $-2x^{2}-3y^{2}+3$. Since we wish to maximize the function, we want to follow these blue vectors.
The red/brown gradient vectors, indicate the direction to move, if you want to increase the value of $x+y$. Since the constraint requires $x+y$ to be constant, we cannot follow these red/brown vectors.
Understanding Lagrange Multiplier Method
Now, when you use Lagrange Multiplier method, you will see that the blue and red/brown vector align at the maximum point location.
An interpretation is: at the maxima, you can no longer increase the value (by following blue vector) without violating the constraint (by following red/brown vector).
Remarks
The gradient vector should lie on the $x-y$ plane. Here is a picture of the contour line of the objective function (the ellipses) and the constraint (the line).