Question

daca a,b,c ∈ R calculati valoarea minima a expresiei

Answer 1

[tex]E=\log_a{bc}+\log_b{ac}+\log_b{ac}\\ \text{Dupa ce schimbam in aceeasi baza(10) se obtine:}\\ E=\dfrac{\lg {bc}}{\lg a}+\dfrac{\lg {ac}}{\lg b}+\dfrac{\lg {ba}}{\lg c}\\ E=\dfrac{\lg b+\lg c}{\lg a}+\dfrac{\lg a+\lg c}{\lg b}+\dfrac{\lg a+\lg b}{\lg c}\\ E=\left(\dfrac{\lg a}{\lg b}+\dfrac{\lg b}{\lg a}\right)+\left(\dfrac{\lg a}{\lg c}+\dfrac{\lg c}{\lg a}\right)+\left(\dfrac{\lg c}{\lg b}+\dfrac{\lg b}{\lg c}\right)\\ \text{Deoarece:} \dfrac{x}{y}+\dfrac{y}{x}\geq 2,\forall x\in \matbb{R_+}\\[/tex]
[tex]Deci: E\geq 2+2+2=6\\ \text{Valoarea minima este 6.}[/tex]