Using the relation `A.M. >= G.M.` prove that
(i)    `(x^2y+y^2z+z^2x)(xy^2+yz^2+zx^2)>=9x^2y^2z^2`, (`x`,`y`,`z` are positive real numbers.)
(ii)    `(a+b)*(b+c)*(c+a)>abc`, (`a`,`b`,`c` are positive real numbers.)

Using the relation A.M>=G.M prove that (x^(2)y+y^(2)z+z^(2)x)(xy^(2)+yz^(2)+zx^(2))>=9(x^(2)y^(2)z^(2))

Using the relation A.M ge G.M . Prove that (i) tan theta +cot theta ge 2 , if 0 lt theta lt (pi)/(2) (ii) (x^2y+y^2z+z^x)(xy^2+yz^2+zx^2) gt 9x^2y^2z^2 . Where x,yz are different real no. (iii) (a+b).(b+c).(c+a) ge 8abc , if a,b,c are positive real numbers.

If x.y,z are positive real numbers such that x^(2)+y^(2)+z^(2)=27, then x^(3)+y^(3)+z^(3) has

Using the properties of determinants, show that: abs((x,x^2,yz),(y,y^2,xz),(z,z^2,xy))=(x−y)(y−z)(z−x)(xy+yz+zx)

Prove that |(x,x^2,yz),(y,y^2,zx),(z,z^2,xy)|= (x-y)(y-z)(z-x)(xy + yz + zx) .

prove that: |(y^(2)z^(2),yz,y+z),(z^(2)x^(2),zx,z+x),(x^(2)y^(2),xy,x+y)|=0

prove that: |(y^(2)z^(2),yz,y+z),(z^(2)x^(2),zx,z+x),(x^(2)y^(2),xy,x+y)|=0

Prove that |[x,y,z] , [x^2, y^2, z^2] , [yz, zx, xy]| = |[1,1,1] , [x^2, y^2, z^2] , [x^3, y^3, z^3]|

If x,y,z are positive real numbers such that x^(2)+y^(2)+Z^(2)=7 and xy+yz+xz=4 then the minimum value of xy is

If x,y,z are positive real numbers such that x^(2)+y^(2)+Z^(2)=7 and xy+yz+xz=4 then the minimum value of xy is