若x>0,y>0,x+y=1则(x+1/x)(y+1/y)的最小值为多少? 写出详细答案
来源:学生作业帮助网 编辑:作业帮 时间:2024/06/17 01:05:44
若x>0,y>0,x+y=1则(x+1/x)(y+1/y)的最小值为多少? 写出详细答案
(x+1/x)(y+1/y)=[(x^2+1)/x][(y^2+1)/y]
=(x^2+y^2+x^2*y^2+1)/xy
=x/y+y/x+xy+1/xy (xy+1/xy不能用均值定理)
=x/y+y/x+xy+(x+y)^2/xy
=2(x/y+y/x)+xy+2 (1=x+y≥2√xy),xy≤1/4,)
≥6+xy=6.25
(x+1/x)(y+1/y)=[(x^2+1)/x][(y^2+1)/y]
=(x^2+y^2+x^2*y^2+1)/xy
=[(x+y)^2-2xy+(xy)^2+1]/xy
=[2-2xy+(xy)^2]/xy=2/xy+xy-2.
设t=xy≤[(x+y)/2]^2=1/4.
f(t)=2/t+t在(0,√2)单减,在(√2,+∞)单增.f(t)=2/t+t在t=1/4时取得最小值.
将式子乘上两个(x+y)因为x+y=1所以原式不变 然后就得到x^2+y^2+2xy+2+x/y+y/x+2=(x+y)^2+2+x/y+y/x=1+2+x/y+y/x然后式子大于等于3+2=5然后最小值是5
(x+1/x)(y+1/y)=(x^2+y^2+x^2*y^2+1)/(xy) (1)
由(x+y)=1,平方得x^2+2xy+y^2=1代入(1)式得
(x^2*y^2-2xy+2)/(xy),换元t=xy得
y=t+2/t+2 其中t=x(1-x),由二次函数的值域得0
原式=6.25
(x+1/x)(y+1/y)=(x+y)(x+1/x)(y+1/y)
若x+y=1,x不等于0则x+(2xy+y^2)/x/(x+y)/x=
若2x-3y+4=0则x(x*x-1)+x(5-x*x)-6y+7
证明:若x>0,y>0,x+y=1,则(x+1/x)(y+1/y)≥25/4
若x,y满足|2x-y-3|+|3x+2y+1|=0,则x= y=
若x,y满足|x-y+1|+(x+y+3)^2=0则x^2-y^2=
若|x+y+1|+(x-2y)的平方=0,则x+2y=?
若|x+y-5|+(x-y+1)²=0,则x²-y²=
若(x+y)(1-x-y)+6=0,则x+y的值是_____
若y=x(x-1)(x-2)(x-3),则y′(0)=
若实数x,y满足x-y+1》=0,x+y>=0,x
若实数x,y满足x-y+1>=0,x+y>=0,x
已知x属于R,y>0,集合A={x|x*x+x+1,-x,-x-1},集合B={-y,-y/2,y+1}若A=B,则x*x+y*y的值为
若|x+y-1|+(x-y-2)²=0,求代数式(x+2y)(x-2y)-(2x-y)(-y-2x)的值.
已知4x+6y=1 若x=y 则x,y的值等于?若x+y=0 则x=?y=?
若实数x,y满足x|x|+y|y|=1,则点(x,y)到直线x+y=0的距离的取值范围是
RT.若x>1,y>0,且满足xy=x^y,x/y=x^3y,则x+y的值为?
若X>1,Y>0,且满足XY=X的Y次方,X/Y=X的3Y次方,则X+Y等于?
若x>1,y>0,且满足 xy=x^y,x/y大于等于x^3y,则x+y的值为