M < f (x) is always true, and the minimum value of m < f (x) is enough.

M > f (x) has a solution and m > f (x) has a minimum value.

M < f (x) has a solution as long as m < f (x) is the maximum.

Note: f (x) > g (x) is always true or has a solution, but it does not meet the above conditions. Please analyze the specific problems.

The reason is that when f(x) takes the maximum value, g(x) does not necessarily take the maximum value at the same time.