True or false: If a function is convex and bounded above, then is constant.
According to...
Convex function - Wikipedia, the free encyclopedia
... a constant can't be a convex function ...
Kind regards
I believe this statement is TRUE, and the squeeze theorem for limits may be helpful to prove it.
Known the minimum point , i.e. . This minimum point is global and there is only one such point.
Choose an arbitrary point . Without loss of generality, assume , i.e. on the right hand side of the minimum point.
Construct a line segment connecting the arbitrary point to the minimum point .
The segment of function should lie between line and line .
Therefore,
(1)
Inequality (1) holds for any and as long as
For an arbitrary , inequality (1) holds as , but
(2)
Remembering both and are bounded.
Thefore,
Here is another method to prove this statement. This method investigates intervals , which differs from previous method which investigates interval .
Choose two arbitrary points and , where .
(1) If , the slope of the straight line is positive. And is above this line in the interval , which implies as , and is unbounded.
(2) If , the slope of the straight line is negative. And is above this line in the interval , which implies as , and is unbounded.
Therefore, we must have , meaning is constant.