Answer:
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.
[tex]\large f:\mathbb{R}\rightarrow \mathbb{R}\;given\;by\;f(x)=x^3[/tex]
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 [tex]\large \mathbb{R}[/tex] is open, but we can also fin a proper open subset and the same holds
[tex]\large f:(-1,1)\rightarrow \mathbb{R}\;,f(x)=x^3[/tex]
c)
A global minimum or global maximum value NOT necessarily exists.
Example of a function that has no global extremum.
[tex]\large g:\mathbb{R}^2\rightarrow \mathbb{R};,g(x,y)=xy[/tex]
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 [tex]\large \mathbb{R}^n[/tex] (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 [tex]\large \mathbb{R}[/tex])
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 [tex]\large \triangledown f[/tex] vanishes (critical points), that is to say, find the points where
[tex]\large \triangledown f=(\frac{\partial f}{\partial x_1},\frac{\partial f}{\partial x_2},...,\frac{\partial f}{\partial x_n})=\bf \bar 0\;the\;zero\;vector[/tex]
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.
