怎么推导?要推导不是证明若a,b∈R+,则x^2/a+y^2/b≥(x+y)^2/a+b⑦(当且仅当x/a=y/b时取等号);

Question

怎么推导?要推导 不是证明 若a,b∈R+,则x^2/a+y^2/b≥(x+y)^2/a+b⑦(当且仅当x/a=y/b时取等号);

Answer 1

y^2* a/b+x^2*b/a>=2√y^2* a/b * x^2*b/axy =2xy
两边各加 x^2+y^2 =>x^2+y^2 +y^2* a/b+x^2*b/a>= x^2+y^2+2xy
=>x^2(1+b/a)+y^2(1+a/b) >= (x+y)^2
=>x^2(a+b)/a +y^2(b+a)/b >= (x+y)^2
两边各除以 (a+b) =>x^2/a+y^2/b≥(x+y)^2/a+b
"="成立时 y^2* a/b=x^2*b/a =>y^2a^2=x^2b^2 => ya=xb => x/a=y/b