費馬：內部極值是駐點
費馬定理說：若 f 在內部點 c 取得局部極大或極小，且 f 在 c 處可微，則 f′(c) = 0。證明從兩側讀取差商的符號。在局部極大處，分子 f(x) − f(c) 在兩側都 ≤ 0；分母變號，於是兩個單側極限把 f′(c) 夾在 ≤ 0 與 ≥ 0 之間。
Fermat: f has a local max at interior c, f differentiable at c => f'(c) = 0.
Local max: f(x) <= f(c) for all x near c, so f(x) - f(c) <= 0 near c.
Approach from the RIGHT (x > c, so x - c > 0):
(f(x) - f(c)) / (x - c) = (<=0) / (>0) <= 0,
hence the right-hand limit f'(c) <= 0.
Approach from the LEFT (x < c, so x - c < 0):
(f(x) - f(c)) / (x - c) = (<=0) / (<0) >= 0,
hence the left-hand limit f'(c) >= 0.
f is differentiable at c, so both one-sided limits equal f'(c). Thus
f'(c) <= 0 AND f'(c) >= 0 => f'(c) = 0. ∎羅爾，然後是中值定理
羅爾定理：若 f 在 [a, b] 上連續、在 (a, b) 上可微，且 f(a) = f(b)，則存在內部點 c 使 f′(c) = 0。證明把兩位先前的巨人結合起來——極值定理給出最大值與最小值，費馬收尾內部情形。
- 由極值定理，連續的 f 在緊緻的 [a, b] 上取得最大值 M 與最小值 m。
- 若兩者都只在端點處取得，則 M = m（因 f(a) = f(b)），於是 f 是常數，處處 f′ = 0——證畢。
- 否則某個極值在內部點 c 取得。在那裡用費馬定理：f′(c) = 0。
中值定理就是把羅爾定理斜著看。減去那條割線，讓端點拉平，用羅爾定理，再斜回去。
MVT: f continuous on [a,b], differentiable on (a,b)
=> exists c in (a,b) with f'(c) = ( f(b) - f(a) ) / ( b - a ).
Define the 'subtract the secant' auxiliary function:
g(x) = f(x) - [ f(a) + (f(b)-f(a))/(b-a) · (x - a) ].
Check g's endpoints:
g(a) = f(a) - f(a) = 0,
g(b) = f(b) - [ f(a) + (f(b)-f(a)) ] = f(b) - f(b) = 0.
So g(a) = g(b) = 0, and g is continuous on [a,b], differentiable on (a,b).
By Rolle, exists c in (a,b) with g'(c) = 0. But
g'(x) = f'(x) - (f(b)-f(a))/(b-a),
so g'(c) = 0 gives f'(c) = ( f(b) - f(a) ) / ( b - a ). ∎