Question

How do I calculate the upper bound of a rectangle?

Answer 1

I will assume that you mean "How do I calculate the upper bound of a function over a rectangle?".

Explanation:

I am not sure what you mean by the question, but I will answer the question: Given a continuous differentiable function `f:RR^2 -> RR`, how do you find the upper bound of the value of `f` over a rectangle `[x_1, x_2] xx [y_1, y_2]`.

The upper bound will be the maximum value of the function, occurring at one of the following locations:

(1) At a corner of the rectangle, i.e. `(x_1, y_1)`, `(x_1, y_2)`, `(x_2, y_1)` or `(x_2, y_2)`.

(2) Along a horizontal edge, where the partial derivative is zero:

`del/(del x) f(x, y_1) = 0` or `del/(del x) f(x, y_2) = 0`

and `x in (x_1, x_2)`

(3) Along a vertical edge, where the partial derivative is zero:

`del/(del y) f(x_1, y) = 0` or `del/(del y) f(x_2, y) = 0`

and `y in (y_1, y_2)`

(4) Inside the body of the rectangle at a point where both partial derivatives are zero:

`del/(del x) f(x, y) = del/(del y) f(x, y) = 0`

and `(x, y) in (x_1, x_2) xx (y_1, y_2)`

Evaluate `f(x, y)` at each of these possible locations and pick the maximum value found.