费马：内部极值是驻点
费马定理说：若 f 在内部点 c 取得局部极大或极小，且 f 在 c 处可微，则 f′(c) = 0。证明从两侧读取差商的符号。在局部极大处，分子 f(x) − f(c) 在两侧都 ≤ 0；分母变号，于是两个单侧极限把 f′(c) 夹在 ≤ 0 与 ≥ 0 之间。
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. ∎罗尔，然后是中值定理
罗尔定理：若 f 在 [a, b] 上连续、在 (a, b) 上可微，且 f(a) = f(b)，则存在内部点 c 使 f′(c) = 0。证明把两位先前的巨人结合起来——极值定理给出最大值与最小值，费马收尾内部情形。
- 由极值定理，连续的 f 在紧致的 [a, b] 上取得最大值 M 与最小值 m。
- 若两者都只在端点处取得，则 M = m（因 f(a) = f(b)），于是 f 是常数，处处 f′ = 0——证毕。
- 否则某个极值在内部点 c 取得。在那里用费马定理：f′(c) = 0。
中值定理就是把罗尔定理斜着看。减去那条割线，让端点拉平，用罗尔定理，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 ). ∎