# Unit 6: Mean Value Theorems (MT242) ## Rolle's Theorem - **Statement**: if $f$ is continuous on $[a,b]$, differentiable on $(a,b)$ and $f(a)=f(b)$, then there exists $c\in(a,b)$ with $f'(c)=0$. - Proof: by the extreme value theorem $f$ attains its maximum and minimum; if both are at endpoints $f$ is constant; otherwise an extremum is interior, so the interior extremum theorem applies. - Geometric meaning: some tangent is horizontal. - Conditions are essential: $f(x)=|x|$ on $[-1,1]$ fails differentiability; $f(x)=x$ on $[0,1)$ fails. - Applications: **between two zeros of a differentiable function lies a zero of $f'$**; number of real roots of an equation; $x^3+px+q=0$ has at most one real root if $p>0$. ## Mean Value Theorem (Lagrange) - **Statement**: if $f$ is continuous on $[a,b]$ and differentiable on $(a,b)$ then there is $c\in(a,b)$ with $$f'(c)=\frac{f(b)-f(a)}{b-a}.$$ - Proof: apply Rolle to $\varphi(x)=f(x)-f(a)-\dfrac{f(b)-f(a)}{b-a}(x-a)$. - Geometric meaning: some tangent is parallel to the chord. - Corollaries: $f'\equiv0$ on an interval implies $f$ is constant; $f'=g'$ implies $f-g$ constant; $f'>0$ implies **increasing**, $f'<0$ **decreasing**; $|f'|\le K$ implies $f$ is Lipschitz. - Another form: $f(a+h)=f(a)+hf'(a+\theta h)$, $0<\theta<1$. ## Intermediate Value Property of Derivatives and Darboux's Theorem - **Darboux's theorem**: if $f$ is differentiable on $[a,b]$ and $k$ lies strictly between $f'(a)$ and $f'(b)$ then there is $c\in(a,b)$ with $f'(c)=k$. - Hence a derivative has the **intermediate value property** even if $f'$ is not continuous. - Proof: for $g(x)=f(x)-kx$, $g'(a)$ and $g'(b)$ have opposite signs; $g$ attains a minimum on $[a,b]$ at an interior point, where $g'=0$. - Consequences: a derivative cannot have a **jump discontinuity**; $f'(x)=\operatorname{sgn}x$ cannot be a derivative of any function on an interval containing 0. ## Applications of the Mean Value Theorem to Inequalities - $|\sin x-\sin y|\le|x-y|$; $|\cos x-\cos y|\le|x-y|$. - $e^x>1+x$ for $x\ne0$; $\dfrac x{1+x}<\ln(1+x)0$. - $\dfrac{b-a}{1+b^2}<\tan^{-1}b-\tan^{-1}a<\dfrac{b-a}{1+a^2}$ for $0-1$, $n\ge1$. - $x>\sin x$ for $x>0$; $\dfrac{\sin x}{x}$ decreasing on $(0,\pi)$. - Method: choose $f$ and the interval, compute $f'(c)$ as the slope, and bound $f'(c)$ using the range of $c$. - Example: for $00$ on an interval implies $f$ is increasing, and use the MVT to show $|\sin x-\sin y|\le|x-y|$. (long) 6. Prove Darboux's theorem. (long)