Informaţii, definiţii, teoreme, formule, exerciţii şi probleme rezolvate din matematica de liceu. RSS/XML

Data publicarii: 16 August, 2010

EXEMPLUL 1

Suport teoretic:

Functia logaritmica, proprietatile logaritmilor, valoarea minima a unei functii.

Enunt:

Sa se determine numarul pozitiv x pentru care functia f are valoarea minima:

$f:{(0,\infty)}\rightarrow{\mathbb{R}},\;f(x)={{{log}_3}^4}x+{8{{{log_3}}^2}x}\cdot{{log}_3}9x.$ f:{(0,\infty)}\rightarrow{\mathbb{R}},\;f(x)={{{log}_3}^4}x+{8{{{log_3}}^2}x}\cdot{{log}_3}9x.

Raspuns:

x = 1.

Rezolvare:

Avem, succesiv:

$f(x)={{{log}_3}^4}x+{8{{{log_3}}^2}x}\cdot{{log}_3}9x={{{log}_3}^4}x+{8{{{log_3}}^2}x}\cdot{({log}_3{x}+{log}_3{9})}=\cdots=$ f(x)={{{log}_3}^4}x+{8{{{log_3}}^2}x}\cdot{{log}_3}9x={{{log}_3}^4}x+{8{{{log_3}}^2}x}\cdot{({log}_3{x}+{log}_3{9})}=\cdots= $={{[{({log}_3{x}+2)}^2-4]}^2}\ge{0},\;\forall{x\in{(0,\;\infty)}}.$ ={{[{({log}_3{x}+2)}^2-4]}^2}\ge{0},\;\forall{x\in{(0,\;\infty)}}.