Suppose you are asked to find minimum and maximum values of a continuous function f: r n → r on one of the following regions: (a) all of r n , (b) an open region in r n (one example is region x 2 1 + . . + x 2 n < 1), (c) a hypersurface in r n comprised of those points satisfying a single constraint equation g(x1, . . , xn) = c (examples include the hypersurface defined by x 2 1 + . . + x 2 n = 1 or x1 + . . + xn = 0), or (d) a closed and bounded region in r n (an example is the region defined by x 2 1 + . . + x 2 n ≤ 1). for each of the four types of regions described above: (1) state and defend whether or not a global minimum or global maximum value necessarily exists (either by quoting a theorem from the text that global extrema exist or by giving a specific example of a function that has no global extremum on a particular region type), and (2) explain, in your own words, the procedure for finding the global extrema in the case(s) where they do exist. (definitions, extrema, regions, procedures)

See examples below

Step-by-step explanation:

a)

A global minimum or global maximum value NOT necessarily exists

Example of a function that has no global extremum.

b)

A global minimum or global maximum value NOT necessarily exists.

Example of a function that has no global extremum.

The same example in a) would work since is open, but we can also fin a proper open subset and the same holds

c)

A global minimum or global maximum value NOT necessarily exists.

Example of a function that has no global extremum.

The level curve g(x,y)=1 is the hyperbola

y = 1/x which has no maxima nor minima.

d)

A global minimum or global maximum value necessarily exists.

Because of the theorem that states that a continuous function f defined in a closed and bounded subset C of   (a compact set) always attains a maximum and a minimum at some points of C (f[C] is closed and bounded, so it is compact in )

2)

If f is continuous AND differentiable, then one way of finding the global extrema (if they exist) is searching for the points where the gradient of the function vanishes (critical points), that is to say, find the points where

that would give us local extrema, then by evaluating the function on each point, find out which ones are maximum or minimum.

If f is only continuous AND NOT differentiable, and somehow we can prove there are global extrema, the only way to find them so far is with computer-assisted numerical methods.

b

step-by-step explanation: