5.2 Derivatives and IVT

only works if x is an interior point

Derivative

Then $f$ is differentiable at $x$ if

Say $f^{\prime }(x)=L$

Example

$f:\mathbb{R}\to \mathbb{R}$,

If $x\ne 0$

if $x=0$

Example

$|x|:\mathbb{R}\to \mathbb{R}$ at 0

($h>0⟹1,h<0⟹-1$)

Example

Differentiable but $f^{\prime }$ is not continuous
$f:\mathbb{R}\to \mathbb{R}$

For $x\ne 0,f^{\prime }(x)=2\sin \left(\frac{1}{x}\right)-(x\sin (\frac{1}{x})-\cos (\frac{1}{x})))$

$x_n=\frac{4}{(8n+1)\pi }\to -1$ apparently
$y_n=\frac{4}{(8n+3)\pi }\to 1$ apparently

At $0,f^{\prime }(0)=0$

Lemma: Differentiable then Continuous

If $f:I\to \mathbb{R}$ is differentiable at $x$ then $f$ is continuous at $x$

Proof

Since $f$ is diff. at $c$, $\underset{x\to c}{lim}\frac{f(x)-f(c)}{x-c}=f^{\prime }(c)$

So it is continuous

Lemma: Differentiable then Lipschitz

If $f:I\to \mathbb{R}$ is differentiable at $x\in I$ then $f$ is locally Lipschitz
$\mathrm{\exists }L>0,\delta >0,|f(x)-f(y)|\le L|x-y|$ for all $y\in I,|x-y|<\delta$

Thm: Rademacher

Lipschitz functions are differentiable everywhere except on a "small set"

Algebraic Properties:

$f,g:I\to \mathbb{R}$ differentiable at $c\in I$
i) $(\lambda f{)}^{\prime }=\lambda f^{\prime }$
ii) $(f+g{)}^{\prime }=f^{\prime }+g^{\prime }$
iii) $(fg{)}^{\prime }=f^{\prime }g+g^{\prime }f$
iv) ${\left(\frac{f}{g}\right)}^{\prime }=\frac{f^{\prime }g-f{g}^{\prime }}{g^2}$
v) $f:J\to \mathbb{R},g:I\to \mathbb{R}$, $g(I)\subset J$ then $(f\circ g{)}^{\prime }=f^{\prime }(g)g^{\prime }$

Thm: Fermat Interior Extremum

$f:(a,b)\to \mathbb{R}$ is diff. at $c\in (a,b)$
If $f$ has a local max or min at $c$ then $f^{\prime }(c)=0$

Proof

Assume $c\in (a,b)$ is a local max.
Let ${\delta }_0=min\{c-a,b-c\}$ and so $c+h\in (a,b)$ for $|h|<{\delta }_0$
Since $c$ is the local max there exists some ${\delta }_1$ so that:

Set $\delta =\frac{1}{2}min\{{\delta }_0,{\delta }_1\}$ For $h>0$ and $|h|<\delta$

Since $f$ is differentiable at $c$

For $h>0$ and $|h|<\delta$

For $h<0$ and $|h|<\delta$

$⟹f^{\prime }(c)=0$

Thm: Darboux

$f$ is differentiable on $[a,b]$ and $f^{\prime }(a)<\alpha <f^{\prime }(b)$ then there exists some $c\in (a,b)$ where $f^{\prime }(c)=\alpha$

Proof

Let $g(x)=f(x)-\alpha x$ on $[a,b]$.
$g$ is differentiable on $[a,b]$ with $g^{\prime }(x)=f^{\prime }(x)-\alpha$. So $g^{\prime }(a)<0<g^{\prime }(b)$.
$g^{\prime }(a)$ and $g^{\prime }(b)$ are not minima and by EVT it must be the case then that there's a $c\in (a,b)$ such that $g^{\prime }(c)=0$

Example

$g:[-1,1],g(x)=\{\begin{array}{ll}1& 0<x\le 1\\ 0& x=0\\ -1& -1\le x<0\end{array}$

Is there a $f^{\prime }=g$? No will violate Darboux

5.3 Mean Value Theorem

Thm: Rolle's Theorem

Let $f:[a,b]\to \mathbb{R}$ be continuous on $[a,b]$ and differentiable on $(a,b)$. If $f(a)=f(b)$ then there exists $c\in (a,b)$ with $f^{\prime }(c)=0$

Proof

since $f$ is continuous on $[a,b]$ then there exists $C_{max},C_{min}\in [a,b]$
Case 1: $C_{max}\in (a,b)$ , $f^{\prime }(C_{max})=0$

Case 2: $C_{min}\in (a,b)$ Set $c=C_{max}$
$f^{\prime }(C_{max})=0$

Case 3: Neither $C_{max}$ or $C_{min}\in [a,b]$, but $f(a)=f(b)$ therefore $f$ is constant on $[a,b]$ so $f^{\prime }=0$

Thm: Mean Value

If $f:[a,b]\to \mathbb{R}$ is continuous on $[a,b]$ and differentiable on $(a,b)$ then there exists a point $c\in (a,b)$ where

Proof

$d$ is continuous on $[a,b]$, differentiable on $(a,b)$ and we have the system of equations:

By Rolle's Theorem there exists a point $c\in (a,b)$ where $d^{\prime }(c)=0$

Thm: Cauchy Mean Value

$f,g\text{:}[a,b]\to \mathbb{R}$ are continuous on $[a,b]$ and differentiable on $(a,b)$. Then there exists $c\in (a,b)$ with

If $g(b)\ne g(a),g^{\prime }(x)\ne 0$, then:

Thm L'Hospitals 0/0

$f,g:(a,b)\to \mathbb{R}$ differentiable and $g(x)$ and $g^{\prime }(x)\ne 0$
For all $x\in (a,b)$. Assume $\underset{x\to a}{lim}f(x)=\underset{x\to a}{lim}g(x)=0$

If $\underset{x\to a}{lim}\frac{f^{\prime }(x)}{g^{\prime }(x)}=L$ then $\underset{x\to a}{lim}\frac{f(x)}{g(x)}=L$

Thm L'Hospitals $\mathrm{\infty }/\mathrm{\infty }$

Assume $f$ and $g$ are differentiable on $(a,b)$ and that $g^{\prime }(x)\ne 0$ for all $x\in (a,b)$.If $\underset{x\to a}{lim}g(x)=\infty$(or $-\infty$), then