# How do I calculate the upper bound of a rectangle?

##### 1 Answer

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

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.

(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