Conditions for a function to have guaranteed global extrema along $[a, b]$ Today, I took my Calculus I final. One of the questions on the final was thus (paraphrased):

Which of the following must be true in
  order for a function to have global
  extrema along the interval $[a, b]$?
  
  
*
  
*The function must be continuous along the interval
  
*The function must be positive along the interval
  
*The function must be increasing along the interval
  
*There is no guarantee of global extrema along the interval
  

(These were with letters, but I can't find a way to duplicate a list with letters easily in markdown.)
For me, numbers two and three were toss-outs. The first seemed correct until I thought of a possible counterexample: $f(x) = k$, where $k$ is some constant.
In this case, it's just a straight line across the interval. There are no global extrema, and the endpoints of the interval aren't extrema either. Which is the correct answer? The one that was on the key was the first, which I did mark by a stroke of luck (she said that 21 of the answers were A and I had 20).
Is this simply a matter of semantics? Or did the creators of this exam simply not consider the possibility of a horizontal line? Or something else?

If the phrasing was : 

Which of the following must occur for a function to have a global extrema in the interval [a,b]

Then we can construct simple counter-examples to the first 3. For example, suppose that the function was identically 0 at every point except at the point $\frac{b+a}{2}$, where it was 1. Clearly, the function has a global maximum there, and the function is not continuous. Thus continuity is not required.
For number 2, consider the parabola $y = -(x - \frac{b+a}{2})^2$ - 1. Everywhere, this function is negative, but it also has a global maximum at the midpoint of a and b.
For number 3, either of the previous two functions also serves as a counter-example.
So if that was the phrasing of the question, then number 4 is correct. But I think that the question might have been phrased differently - it might have asked which of the 4 guarantees that there is a global maximum. In this case, continuity guarantees both a maximum and a minimum, as seen here.
