Question

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)

Answer 1

See examples below

Explanation:

a)

A global minimum or global maximum value NOT necessarily exists

Example of a function that has no global extremum.

`\large f:\mathbb{R}\rightarrow \mathbb{R}\;given\;by\;f(x)=x^3`

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
`\large \mathbb{R}` is open, but we can also fin a proper open subset and the same holds

`\large f:(-1,1)\rightarrow \mathbb{R}\;,f(x)=x^3`

c)

A global minimum or global maximum value NOT necessarily exists.

Example of a function that has no global extremum.

`\large g:\mathbb{R}^2\rightarrow \mathbb{R};,g(x,y)=xy`

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
`\large \mathbb{R}^n` (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
`\large \mathbb{R}`)

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
`\large \triangledown f` vanishes (critical points), that is to say, find the points where

`\large \triangledown f=((\partial f)/(\partial x_1),(\partial f)/(\partial x_2),...,(\partial f)/(\partial x_n))=\bf \bar 0\;the\;zero\;vector`

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.