f′ 的符號控制單調性
這是中值定理的第一份紅利：在一個區間上 f′ > 0 意味著 f 在那裡嚴格遞增。這就是遞增判別法，儘管它看起來理所當然，卻真的需要中值定理——導數是局部量，遞增是整體陳述，只有中值定理把二者連起來。
Claim: if f' > 0 on an interval I, then f is strictly increasing on I.
Proof. Take any x1 < x2 in I. Apply the MVT to f on [x1, x2]:
there is c in (x1, x2) with
f(x2) - f(x1) = f'(c) · (x2 - x1).
Now f'(c) > 0 (given) and x2 - x1 > 0, so the right side is > 0.
Hence f(x2) - f(x1) > 0, i.e. f(x2) > f(x1).
Since x1 < x2 were arbitrary, f is strictly increasing on I. ∎
Corollary (constancy): if f' = 0 on I, the same identity gives
f(x2) - f(x1) = 0 for all x1, x2, so f is constant on I.柯西中值定理與羅必達
柯西中值定理同時處理兩個函數：在通常假設下，存在 c 使 (f(b) − f(a)) g′(c) = (g(b) − g(a)) f′(c)。證明是對巧妙的組合 h(x) = (f(b) − f(a)) g(x) − (g(b) − g(a)) f(x) 用羅爾定理，它的端點相等。這種雙函數形式，正是羅必達法則背後的引擎。
L'Hopital (0/0 form): suppose f, g differentiable near a, f(a) = g(a) = 0,
g'(x) ≠ 0 near a, and lim_{x->a} f'(x)/g'(x) = L. Then lim_{x->a} f(x)/g(x) = L.
Why, via Cauchy MVT. For x near a (say x > a), apply the Cauchy MVT on [a, x]:
there is c between a and x with
( f(x) - f(a) ) / ( g(x) - g(a) ) = f'(c) / g'(c).
But f(a) = g(a) = 0, so the left side is just f(x)/g(x). Hence
f(x) / g(x) = f'(c) / g'(c), with a < c < x.
As x -> a, the squeezed c -> a too, so f'(c)/g'(c) -> L.
Therefore f(x)/g(x) -> L. ∎
Worked instance: lim_{x->0} (sin x)/x. Here f=sin x, g=x, f(0)=g(0)=0,
f'/g' = (cos x)/1 -> cos 0 = 1. So the limit is 1 -- recovered honestly.