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

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 properties of determinants , prove that |(x^2+1,xy,zx),(xy,y^2+1,yz),(zx,yz,z^2+1)|=1+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.

Using properties of determinants, prove that |{:(y^2z^2,yz,y+z),(z^2x^2,zx,z+x),(x^2y^2,xy,x+y):}|=0

Using the proprties of determinants in Exercise 7 to 9, prove that |{:(y^2z^2,yz,y+z),(x^2z^2,zx,z+x),(x^2y^2,xy,x+y):}|=0

Verify that: (x+y+z)^(2)=x^(2)+y^(2)+z^(2)+2xy+2yz+2zx

If x+y+2z=0 then prove that x^2/(yz)+y^2/(zx)+(8z^2)/(xy)=6

Factorise: x^2+y^2+z^2-xy-yz-zx