\subsubsection {}
Doka\v zte, \v ze funkce $f$ je na intervalu $I$
rostouc\'\i{}, pr\'av\v e kdy\v z plat\'\i{}
$$\forall_{x,y \in I, x \ne y} \frac{f(x)-f(y)}{x-y}>0.$$
 



\subsubsection {}
Je-li $f$ neklesaj\'\i c\'\i {} na $(-\infty,a>$ a
nerostouc\'\i {} na $<a,\infty)$ pro n\v ejak\'e $a \in \Bbb
R$, pak $f$ m\'a maximum.
 



\subsubsection {}
Necht' $f$ m\'a minimum v $a\in {\Bbb R }$. Mus\'\i {}
pak existovat $\epsilon>0$ takov\'e, \v ze $f$ je
nerostouc\'\i {} na $(a-\epsilon,a>$ a neklesaj\'\i c\'\i {}
na $<a,a+\epsilon)$?
 



\subsubsection {}
Kter\'e z n\'asleduj\'\i c\'\i ch operac\'\i {}
proveden\'ych na neklesaj\'\i c\'\i {} (monot\'on-n\'\i {}) funkce d\'avaj\'\i {}
op\v et neklesaj\'\i c\'\i {} (monot\'onn\'\i {}) funkci?

Operace: $+, -, max, min, \circ.$

Existuje funkce, kter\'a nen\'\i {} monot\'onn\'\i {} na \v
z\'adn\'em intervalu?
 



\subsubsection {}
Existuje funkce $f:{\Bbb R }\to {\Bbb R }$, kter� nem� v ��dn�m bod� extr�m?
\smallskip
 



\subsubsection {}
Studujte vlastnosti a nakreslete grafy
n\'asleduj\'\i c\'\i ch funkc\'\i {}:

$$f_1(x)=\frac{2x}{x^2+1}, f_2(x)=x^n (n\in{\Bbb N }),
f_3(x)=\frac{5-|x-7|}{2},$$
$$f_4(x)=(x-5)^2+6, f_5(x)=x^2+2x-7, f_6(x)=\frac{-1}{|x-3|}.$$
 



\subsubsection {}
Rozhodn\v ete, kter\'a z n\'asleduj\'\i c\'\i ch
func\'\i {} je polynomi\'aln\'\i {}.
$$f(x)=\frac{x^4+2x^2+1}{(\sqrt{x^2+2}-1)(\sqrt{x^2+2}+1)},
g(x)=\sqrt{x^2+1}.$$


\subsubsection {}
P��klady na pr�b�h funkc� 

1. $f(x)={2x\over 1-x^2}$ 

2. $f(x)=x\sqrt{1-x^2}$

3. $f(x)=\root 3 \of {x^3-x^2-x+1}$

4. $f(x)=\root3 \of {x^2}-{2\over\sqrt x}$ 

5. $f(x)=\root3\of{x^2}-\root3\of{x^2+1}$

6.  $f(x)={x^4+8\over x^3+1}$

7. $f(x)=(x+1)^{\frac23}+(x-1)^{\frac23}$ 

8. $f(x)={\vert x+1 \vert^{\frac32}\over\sqrt x}$

9. $f(x)=1-x+\sqrt{x^3\over 3+x}$

10. $f(x)=\sin x+\cos^2 x$ 

11.  $f(x)=\vert\sin x\vert +\cos 2x$

12. $f(x)=\exp(-x^2+3x-7)$



\subsubsection {}
Dal�� p��klady na pr�b�h funkc�

1 $f(x)=\operatorname{arctg} {x+1\over x-1}$ 

2. $f(x)=\arcsin {x^2-1\over x^2+1}$ 

3. $f(x)=\arccos {2x\over 1+x^2}$

4. $f(x)=\arcsin(\cos^2 x)$ 

5.  $f(x)={\arccos x\over\sqrt{1-x^2}}$ 

*6. $f(x)={\arccos x\over\sqrt{1-x}}$

*7. $f(x)=\root 3 \of {\frac2\pi \operatorname{arctg}{x\over x^2-1}}$ 

*8. $f(x)=\arcsin(\frac4\pi\operatorname{arctg}\sqrt{1-x^2})$

*9. $f(x)=\vert\operatorname{arctg}{1-x\sqrt3\over x-\sqrt3}-\frac\pi4 \vert$





 

