Math Help - max min on open closed intervals

1. max min on open closed intervals

f is conrinues on (a,b)
if f(a)=f(b) and min(f[a,b])<max(f[a,b])
then we have another extreme point on (a,b)
why?

2. Originally Posted by transgalactic
f is conrinues on (a,b)
if f(a)=f(b) and min(f[a,b])<max(f[a,b])
then we have another extreme point on (a,b)
another extreme point on (a,b), What does that mean exactly?

3. we have minimum or maximum on (a,b)

4. .

5. If f(a)= f(b) and min f[a, b}< max f[a,b], then there are three possiblities.
1) The minimum value of f may occur at the endpoints. Since the maximum is different, it cannot occur at either endpoint and so must occur in the interior.

2) The maximum value of f may occur at the endpoints. Since the minimum is different, it cannot occur at either endpoint and so must occur in the interior.

3) Neither maximum nor minimum occurs at the endpoints. In that case they both occur in the interior.

In any case, there is at least one extremum in the interior of the interval, (a, b).