Question

subpunctele a si b va rog frumos dau coroana si multe puncte

Answer 1

Explicație pas cu pas:

a)

pentru ab > 0

$\dfrac{a}{b} + \dfrac{b}{a} = \dfrac{ {a}^{2} + {b}^{2} }{ab} = \dfrac{ {a}^{2} + {b}^{2} - 2ab + 2ab }{ab} = \dfrac{ {(a - b)}^{2}}{ab} + 2 > 0$

$\iff \dfrac{a}{b} + \dfrac{b}{a} > 0$

b) inegalitatea mediilor:

$\dfrac{a + b}{2} \geqslant \sqrt{ab} \iff a + b \geqslant 2 \sqrt{ab}$

$\dfrac{b + c}{2} \geqslant \sqrt{bc} \iff b + c \geqslant 2 \sqrt{bc}$

$\dfrac{c + a}{2} \geqslant \sqrt{ca} \iff c + a \geqslant 2 \sqrt{ca}$

înmulțim:

$(a + b)(b + c)(c + a) \geqslant {2}^{3} \sqrt{ {a}^{2} {b}^{2} {c}^{2} }$

$(a + b)(b + c)(c + a) \geqslant 8 abc$

q.e.d.