Monotonost funkcije

Arkus kosinus, monotono opadajuća funkcija

Monotonost funkcije u matematici označava njeno svojstvo da joj se sa porastom vrednosti argumenta vrednosti nižu od većih ka manjim ili od manjih ka većim. Grafički predstavljeno, ovo daje liniju koja izgleda kao uzlazna ili silazna linija, a funkcija sa ovim svojstvom se zove monotona funkcija.

Vrednosti ne moraju biti striktno uzlazne, tj. dozvoljena je pojava više jednakih vrednosti. U tom slučaju, funkcija bi bila monotno nerastuća ili monotono neopadajuća. Ukoliko do ovoga ne dolazi, radi se o monotono opadajućoj ili monotono rastućoj funkciji.

Definicije

Monotonost funkcije označava svojstvo onih funkcija koje zadovoljavaju bilo koji od sledećih uslova:

$(\forall x_{1},x_{2}\in B)(x_{1}\leq x_{2}\rightarrow f(x_{1})\leq f(x_{2}))$ - rastuća na $B$
$(\forall x_{1},x_{2}\in B)(x_{1}<x_{2}\rightarrow f(x_{1})<f(x_{2}))$ - strogo rastuća na $B$
$(\forall x_{1},x_{2}\in B)(x_{1}\leq x_{2}\rightarrow f(x_{1})\geq f(x_{2}))$ - opadajuća na $B$
$(\forall x_{1},x_{2}\in B)(x_{1}<x_{2}\rightarrow f(x_{1})>f(x_{2}))$ - strogo opadajuća na $B$

Za funkciju koja zadovoljava ovo svojstvo (tj. bilo koje od četiri navedena svojstva) kažemo da je monotona na skupu $B$ . Specijalno, za funkciju koja zadovoljava drugo ili četvrto svojstvo od četiri navedena, kažemo da je strogo monotona na $B$ .

Ako je funkcija f neprekidna na zatvorenom intervalu [a,b], i diferencijabilna na otvorenom intervalu (a,b); onda je:

funkcija f, monotono neopadajuća za svako x iz intervala (a,b), gde je f ' (x) veće ili jednako nuli.
funkcija f, monotono nerastuća za svako x iz intervala (a,b), gde je f ' (x) manje ili jednako nuli.

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