Réponses

2014-09-24T23:25:36+02:00
13.x
a.
(x-y)²≥0 ⇒x²+y²-2xy≥0 ⇒ x²+y²≥2xy
⇒ (x²+y²)/(xy)≥2 ⇒ x/y+y/x≥2
b.
x<y ⇒x²<xy ⇒√x²<√(xy) ⇒x<√(xy)
x<y ⇒ xy<y² ⇒ √(xy)<y
soit x<√(xy)<y
c.
x/y+y/x≥2 ⇒ x/y+1+y/x+1≥4 ⇒(x+y)/y+(x+y)/x≥4>1
⇒(x+y)(1/y+1/x)>1 ⇒1/y+1/x>1/(x+y) ⇒1/(x+y)<1/x+1/y
d.
2√(xy)>0 ⇒(√x)²+2√(xy)+(√y)²>(√x)²+(√y)²
⇒(√x+√y)²>x+y ⇒(√x+√y)²>(√(x+y))² ⇒ √x+√y > √(x+y) ⇒√(x+y)<√x+√y

14.
a. (y-x)(y²+xy+x²)=y³+xy²+x²y-xy²-x²y-x³=y³-x³
b. (y+x/2)²+3x²/4=y²+xy+x²/4+3x²/4=y²+xy+x²
De b. on d'eduit que x²+xy+y²≥0
si x≤y alors y-x≥0 soit y³-x³=(y-x)(x²+xy+y²)≥0
On en deduit aue y³≥x³