SECTION 4.3: Global Maxima and Minima

The technique/ procedure for finding the global maximum or minimum in a graph is known as optimization. Local maxima or minima occur when a function takes larger or smaler values than nearby points.

A function f is said to have a global maximum at a if f(x) is <= all values of f. It is said to have a global maximum at a if f(x)>= all values of f.

To find the global maximum or minimum of a continuous function on an interval from point a to b, one must compare the values of the function at all critical points in the interval as well as the endpoints, i.e., a and b.

To find the global maximum and minimum of a continuous function on an interval excluding endpoints or on the entire real line, one must find the values of the function at al critical points and then sketch a graph through them.

SECTION 9.5: Critical Points and Optimization

Optimization means finding the largest or smallest values of a function. When interpreting graphs, the method of optimization is used to determine local or global maxima and minima.

A function f is said to have a local maximum at P0 if f(P0) >= f(P) for all points P near P0.
It is said to have a local minimum at Po if f(Po) <= f(P) for all points P near P0.

As explained above, the function has a global maximum at P0 if f(P0) >= f(P) for all points P in R and it has a global minimum at P0 if f(P0) <= f(P) at all points P in R.

If a function f(x,y) has a local maximum or minimum at a point (x0, y0) not in the limits of the domain of f, then either fx (x0,y0) = 0 and fy (x0,y0) = 0 or at least one partial derivative is undefined at the points (x0, y0). Points where each of the partial derivatives is either zero or undefined are called critical points. This is how a critical point of a function can be found analytically.

The part of these two reading sections I found difficult to understand was exactly how one can determine whether a critical point is a local maximum or minimum. The example of the function in the textbook was to confusing.

It is evident that understanding how to determine maximum and minimum poitns whether local or global is useful in economics and the business world because companies need to know how to maximize profits and minimize costs. These can in effect be determined by studying trends on a graph.