神马作文网已知x,y∈R+,且x≠y,求证x^5y^-5+x^-5y^5＞x^4y^-4+x^-4y^4
来源：学生作业帮 编辑：神马作文网作业帮 分类：数学作业 时间：2024/11/10 23:52:55
已知x,y∈R+,且x≠y,求证x^5y^-5+x^-5y^5＞x^4y^-4+x^-4y^4
大家帮帮忙!
大家帮帮忙!
作差法：
(x/y)^5+(y/x)^5-(x/y)^4-(y/x)^4
=(x^10+y^10-x^9y-xy^9)/(x^5y^5)
=(x^9-y^9)(x-y)/(x^5y^5)
=(x^3-y^3)(x^6+x^3y^3+y^6)(x-y)/(x^5y^5)
=(x-y)(x^2+xy+y^2)(x^6+x^3y^3+y^6)(x-y)/(x^5y^5)
=(x-y)^2(x^2+xy+y^2)(x^6+x^3y^3+y^6)/(x^5y^5)
因为x,y∈R+,且x≠y
所以(x-y)^2>0,(x^2+xy+y^2)>0,(x^6+x^3y^3+y^6)>0,(x^5y^5)>0
所以(x-y)^2(x^2+xy+y^2)(x^6+x^3y^3+y^6)/(x^5y^5)>0
所以(x/y)^5+(y/x)^5>(x/y)^4+(y/x)^4
(x/y)^5+(y/x)^5-(x/y)^4-(y/x)^4
=(x^10+y^10-x^9y-xy^9)/(x^5y^5)
=(x^9-y^9)(x-y)/(x^5y^5)
=(x^3-y^3)(x^6+x^3y^3+y^6)(x-y)/(x^5y^5)
=(x-y)(x^2+xy+y^2)(x^6+x^3y^3+y^6)(x-y)/(x^5y^5)
=(x-y)^2(x^2+xy+y^2)(x^6+x^3y^3+y^6)/(x^5y^5)
因为x,y∈R+,且x≠y
所以(x-y)^2>0,(x^2+xy+y^2)>0,(x^6+x^3y^3+y^6)>0,(x^5y^5)>0
所以(x-y)^2(x^2+xy+y^2)(x^6+x^3y^3+y^6)/(x^5y^5)>0
所以(x/y)^5+(y/x)^5>(x/y)^4+(y/x)^4
已知：4x = 5y,求(x+y) ：y
已知x*x+4x+y*y-2y+5=0,则x*x+y*y=?
(x+y)²+4(x+y)-5
x,y属于R*,且x+y=1,求证:(1)(x+1/x)(y+1/y)≥25/4 (2)(x+1/x)^2+(y+1/y
1) (2x+5y)(2x-5y)(-4x^2-25y^2) 2) [（x+y)(x-y)-(x-y)^2+2y(x-y
已知|2-x|与|x-y+4|互为相反数,求代数式3(x-y)-5(x-y)-3(x-y)+(y-x)-4(x+y)+3
已知x²+y²+5=2x+4y,求【2x²-(x-y)(x-y)】【(x+y-1)(x-y
已知x=1/3,y=-1/2,求代数式x-(x+y)+(x+2y)-(x+3y)+(x+4y)-(x+5y)+...-(
已知5x-4y=0且xy不等于0,求x^2+y^2/x^2-y^2-x+y/x-y的值
已知x,y为实数,且2（x+2y）＝x²+y²+5求代数式3x-4y分之x²+y²
（5x-3y)(3y+5x)-(4y-3x)(3x-4y)
（5x+3y)(3y-5x)-(4x-y)(4y+x)=