1.已知a*x^3=b*y^3=c*z^3且1/x+1/y+1/z=1求证(a*x^2+b*y^2+c*z^2)^(1/3)=a^(1/3)+b^(1/3)+c^(1/3){提示令a*x^3=b*y^3=c*z^3=T则a*x^2=T/Xb*y^2=T/Yc*z^2=T/Z}2.已知实数a,b满足a-3*a^(2/3)+5*a^(1/3)=5b-3*b^(2/3)+5*b^(1/3)=5求证a^(1/3)+b^

Question

1.已知 a*x^3=b*y^3=c*z^3 且1/x+1/y+1/z=1 求证(a*x^2+b*y^2+c*z^2)^(1/3)=a^(1/3)+b^(1/3)+c^(1/3) {提示令a*x^3=b*y^3=c*z^3=T 则a*x^2=T/X b*y^2=T/Y c*z^2=T/Z}
2.已知实数a,b满足a-3*a^(2/3)+5*a^(1/3)=5 b-3*b^(2/3)+5*b^(1/3)=5 求证a^(1/3)+b^(1/3)=2

Answer 1

因为a*x^2=T/X b*y^2=T/Y c*z^2=T/Z
所以a*x^2+b*y^2+c*z^2=T/X+T/Y+T/Z=T*(1/x+1/y+1/z)=T
所以(a*x^2+b*y^2+c*z^2)^(1/3)=T^(1/3)=x*a^(1/3)=y*b^(1/3)=z*c^(1/3)..(1)
所以T^(1/3)=x*a^(1/3)/3+y*b^(1/3)/3+z*c^(1/3)/3.(2)
而a^(1/3)=x*a^(1/3)/x b^(1/3)=y*b^(1/3)/y c^(1/3)=z*c^(1/3)/z.(3)
综合（1）（2）（3）(a*x^2+b*y^2+c*z^2)^(1/3)-（a^(1/3)+b^(1/3)+c^(1/3)）
=x*a^(1/3)*（1/x-1/3）+y*b^(1/3)*（1/y-1/3）+z*c^(1/3)*(1/z-1/3).(4)
因为x*a^3=（x*a^(1/3)）^3 y*b^(1/3)=……
所以（4）=x*a^3（1/x+1/y+1/z-3*1/3)=0 所以得证
令a^(1/3）＝X,b^(1/3)=Y
则X^3-3X^2+5X=5,Y^3-3Y^2+5Y=5
得到（X^2+5）（X-1)=0
则X=1
同理Y=1
X+Y=2