in all 3 cases, there is at least one point ( c, f(c) ) on the graph of f(x) where the tangent is horizontal, making f'(c) equal to 0
- I. f(x) = k
- y = a constant f'(x) = 0, so c can be any # in (a,b)
- II. f(x) > f(a) for some x in (a,b)
- * because of condition 1 that f(x) is cont on [a,b], EVT can be used/applied & f(x) has a max val in [a,b]
- * since f(a) = f(b), f(x) has to reach this max val @ a #c in (a,b)
- * then f(x) has a local max @ #c &, by condition 2 that f(x) is dx-able on (a,b), f(x) is dx-able at c * therefore, f'(c) = 0 by Fermat's theorem
- III. f(x) < f(a) for some x in (a, b)
- * by EVT, f(x) has a min val in [a,b] since f(a) = f(b), f(x) gets to this min val @ #c in (a,b)
- * f'(c) = 0 by Fermat's theorem