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).