Answer:
See examples below
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.