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Absolute Maxima and Minima of Functions with Two Variables

In a Nut Shell: Absolute maximum and minimum values of functions in the case of two

independent variables, (x,y), occur at the "critical locations" within or on the boundary

of the domain, D.

A local maximum occurs near (a,b) if f(x,y) ≤ f(a,b). Likewise a local minimum

occurs near (a,b) if f(x,y) ≥ f(a,b). An absolute maximum occurs at (a,b) if f(x,y) ≤ f(a,b)

for all points (x,y) in the domain of the function and an absolute minimum occurs at

(a,b) if f(x,y) ≥ f(a,b) for all points (x,y) in the domain of the function.

You must investigate both interior points and points on the boundary of the domain.

Locating critical points of a function, f(x,y on the interior of the domain):

The slope of the function, f(x,y), must be zero at each critical point, (a,b). For functions

of two independent variables, (x,y), the following partial derivatives must hold:

∂f(a,b)/∂x = 0 and ∂f(a,b)/∂y = 0

Strategy for finding Absolute Minimum, Absolute Maximum of f(x,y)

Step 1

Locate critical points, (a,b)

within the domain

∂f(a,b)/∂x = 0 and ∂f(a,b)/∂y = 0

Step 2

Calculate f(a,b) for each

critical point within in

the domain, D.

Locates local maxima and local

Minima.

Step 3

Calculate f(x,y) on the

boundaries of the domain

Find extreme values of f(x,y) on the

boundaries of the domain, D, by

setting the first derivative of the function

along the boundary to zero and

calculating the extreme values at the

critical points on the boundary.

The largest value from step 2 and 3 is the absolute maximum and the smallest value

from step 2 and step 3 is the absolute minimum.

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