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

Sep 20, 2015

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 $f : {\mathbb{R}}^{2} \to \mathbb{R}$, how do you find the upper bound of the value of $f$ over a rectangle $\left[{x}_{1} , {x}_{2}\right] \times \left[{y}_{1} , {y}_{2}\right]$.

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. $\left({x}_{1} , {y}_{1}\right)$, $\left({x}_{1} , {y}_{2}\right)$, $\left({x}_{2} , {y}_{1}\right)$ or $\left({x}_{2} , {y}_{2}\right)$.

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

$\frac{\partial}{\partial x} f \left(x , {y}_{1}\right) = 0$ or $\frac{\partial}{\partial x} f \left(x , {y}_{2}\right) = 0$

and $x \in \left({x}_{1} , {x}_{2}\right)$

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

$\frac{\partial}{\partial y} f \left({x}_{1} , y\right) = 0$ or $\frac{\partial}{\partial y} f \left({x}_{2} , y\right) = 0$

and $y \in \left({y}_{1} , {y}_{2}\right)$

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

$\frac{\partial}{\partial x} f \left(x , y\right) = \frac{\partial}{\partial y} f \left(x , y\right) = 0$

and $\left(x , y\right) \in \left({x}_{1} , {x}_{2}\right) \times \left({y}_{1} , {y}_{2}\right)$

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