Home/ Intrebari/Q 77662
Urmator
In Process

AniDeȘcoală.ro Latest Intrebari

Mihai96
Pe: 2012-08-10T11:29:46+03:00 In: In:

ordonarea nr reale

se apropie scoala si m-am hotarat sa ma apuc de niste exercitii,dar nu gasesc nici o modalitate de rezolvare… cry: , deci va cer ajutorul
„Daca a, b, c nr reale pozitive se se arate ca:

(a*b)/(a+b) + (b*c)/(b+c) + (c*a)/(c+a) <= (a + b + c)/2

vas ruga daca imi puteti spune cateva notiuni care sa ma ajute la acest tip de exercitii

Similare

1 raspuns

  1. GreatMath
    veteran (III)

    \begin{array}{l} \sum {\frac{{ab}}{{a + b}}} \le \frac{{a{\rm{ }} + {\rm{ }}b{\rm{ }} + {\rm{ }}c}}{2}\\ {\left( {a + b} \right)^2} \ge 4ab \Leftrightarrow {a^2} + 2ab + {b^2} \ge 4ab \Leftrightarrow {\left( {a - b} \right)^2} \ge 0 \to {\rm{Adevarata}}{\rm{.}}\\ {\left( {a + b} \right)^2} \ge 4ab \Leftrightarrow \frac{1}{{{{\left( {a + b} \right)}^2}}} \le \frac{1}{{4ab}} \Leftrightarrow \frac{{ab}}{{a + b}} \le \frac{{a + b}}{4}\\ {\rm{Asemanator}}:\\ \frac{{bc}}{{b + c}} \le \frac{{b + c}}{4}\\ \frac{{ca}}{{c + a}} \le \frac{{c + a}}{4}\\ - - - - - - - - - - - \\ \sum {\frac{{ab}}{{a + b}}} = \frac{{ab}}{{a + b}} + \frac{{bc}}{{b + c}} + \frac{{ca}}{{c + a}} \le \frac{{a + b}}{4} + \frac{{b + c}}{4} + \frac{{c + a}}{4} = \frac{{a{\rm{ }} + {\rm{ }}b{\rm{ }} + {\rm{ }}c}}{2} \end{array}

Raspunde

Raspunde