Докажите, пожалуйста, неравенства:
1) [tex] x^{3}+y^{3} \geq x^{2} y+x y^{2} [/tex] (x>0, y>0)
2) [tex]a^{2} + \frac{1}{ a^{2} } \geq 2[/tex]
1) [tex] x^{3}+y^{3} \geq x^{2} y+x y^{2} [/tex] (x>0, y>0)
2) [tex]a^{2} + \frac{1}{ a^{2} } \geq 2[/tex]
Answers & Comments
Verified answer
1(x+y)(x²-xy+y²)-xy(x+y)≥0
(x+y)(x²-xy+y²-xy)≥0
(x+y)(x²-2xy+y²)≥0
(x+y)(x-y)²≥0
x>0 U y>0⇒x+y>0 U (x-y)²≥0⇒
x³+y³≥x²y+xy²
2
a²+1/a²≥2
(a+1/a)²-2≥2
(a+1/a)-4≥0
(a+1/a-2)(a+1/a+2)≥0
(a²-2a+1)/a *(a²+2a+1)/a≥0
(a-1)²(a+1)²/a²≥0
(a-1)²≥0 при лбом а,(а+1)²≥0 при любом а и а²>0 при любом а⇒
a²+1/a²≥2