Mean value theorems turn local derivative information into global finite-difference information. They are the engine behind monotonicity, inverse estimates, and Taylor remainders.
Rolle's theorem
Rolle's theorem is the zero-slope principle between equal endpoint values.
Theorem. If \(f\) is continuous on \([a,b]\), differentiable on \((a,b)\), and \(f(a)=f(b)\), then some \(c\in(a,b)\) satisfies \(f'(c)=0\).
Proof. By compactness, \(f\) attains a maximum and minimum. If both occur only at the endpoints, the function is constant and any interior point works. Otherwise an interior extremum occurs, and the derivative there is zero by comparing difference quotients from the left and right.
Lemma (Fermat). If \(f\) has a local maximum or minimum at an interior point \(c\) and is differentiable there, then \(f'(c)=0\).
Proof. At a local maximum, \[ \frac{f(c+h)-f(c)}h\le0\quad(h>0),\qquad \frac{f(c+h)-f(c)}h\ge0\quad(h<0). \] If the two-sided derivative exists, it is both nonpositive and nonnegative, hence zero. The minimum case follows by applying this to \(-f\).
Lagrange mean value theorem
The average slope is realized somewhere inside the interval.
Theorem. If \(f\) is continuous on \([a,b]\) and differentiable on \((a,b)\), then there is \(c\in(a,b)\) such that \(f(b)-f(a)=f'(c)(b-a)\).
Proof. Apply Rolle's theorem to \(g(x)=f(x)-\ell(x)\), where \(\ell\) is the secant line joining \((a,f(a))\) and \((b,f(b))\). Then \(g(a)=g(b)=0\), so \(g'(c)=0\), which gives the formula.
Cauchy mean value theorem
Cauchy's form compares two functions and underlies l'Hopital-type arguments.
Theorem. If \(f,g\) are continuous on \([a,b]\), differentiable on \((a,b)\), and \(g'\) does not vanish, then for some \(c\in(a,b)\), \[ \frac{f(b)-f(a)}{g(b)-g(a)}=\frac{f'(c)}{g'(c)}. \]
Proof. Since \(g'\) never vanishes, Rolle's theorem implies \(g(b)\ne g(a)\). Apply Rolle's theorem to \(h(x)=(f(b)-f(a))(g(x)-g(a))-(g(b)-g(a))(f(x)-f(a))\). Then \(h(a)=h(b)=0\), and \(h'(c)=0\) gives the identity.
Monotonicity and derivative sign
Derivative sign controls global order.
Proposition. If \(f'\ge0\) on an interval, then \(f\) is increasing. If \(f'=0\) on an interval, then \(f\) is constant.
Proof. For \(x\lt y\), the mean value theorem gives \(f(y)-f(x)=f'(c)(y-x)\) for some \(c\in(x,y)\). The sign of \(f'\) gives the conclusion.
Derivative bounds
Uniform derivative bounds give Lipschitz estimates.
Corollary. If \(|f'|\le M\) on an interval, then \(|f(x)-f(y)|\le M|x-y|\) for all \(x,y\) in the interval.
Proof. Apply the mean value theorem on the interval between \(x\) and \(y\): \(|f(x)-f(y)|=|f'(c)||x-y|\le M|x-y|\).
Corollary (inverse stability). If \(f'\ge m>0\) on an interval \(I\), then \(f\) is injective and \[ |x-y|\le \frac1m|f(x)-f(y)|,\qquad x,y\in I. \] Thus \(f^{-1}\) is \(1/m\)-Lipschitz on \(f(I)\).
Proof. For \(x<y\), the mean value theorem gives \(f(y)-f(x)=f'(c)(y-x)\ge m(y-x)>0\). This proves strict increase and the displayed estimate.
Darboux property of derivatives
A derivative need not be continuous, but it cannot jump over an intermediate value.
Theorem (Darboux). If \(f\) is differentiable on an interval and \[ f'(a)<\lambda<f'(b) \] for \(a<b\), then \(f'(c)=\lambda\) for some \(c\in(a,b)\).
Proof. Put \(g(x)=f(x)-\lambda x\). Then \(g'(a)<0\) and \(g'(b)>0\). On compact \([a,b]\), \(g\) attains a minimum. The derivative signs show that neither endpoint can be a minimizer: \(g'(a)<0\) means \(g\) decreases immediately to the right of \(a\), and \(g'(b)>0\) means values immediately to the left of \(b\) are smaller than \(g(b)\). Hence a minimizer \(c\) lies in \((a,b)\), and Fermat's lemma gives \(0=g'(c)=f'(c)-\lambda\).
L'H?pital rule from Cauchy's theorem
Theorem (one-sided \(0/0\) form). Suppose \(f(x),g(x)\to0\) as \(x\downarrow a\), both are differentiable on \((a,b)\), \(g'\ne0\), and \[ \frac{f'(x)}{g'(x)}\longrightarrow L\in\overline{\mathbb R}. \] Then \(f(x)/g(x)\to L\), wherever the quotient is defined.
Proof. Extend \(f,g\) continuously to \(a\) by zero. For each \(x>a\), Cauchy's mean value theorem on \([a,x]\) gives a point \(c_x\in(a,x)\) such that \[ \frac{f(x)}{g(x)}=\frac{f'(c_x)}{g'(c_x)}. \] As \(x\downarrow a\), also \(c_x\downarrow a\), and the assumed derivative-ratio limit gives the result. The hypotheses matter: existence of the derivative ratio only along selected sequences is not enough.