Prove that `[(x^2+y^2+z^2)/(x+y+z)]^(x+y+z)> x^x y^y z^z >[(x+y+z)/3]^(x+y+z)(x ,y ,z >0)`

A person travels three equal distances at a speed of x km/hr, y km/hr and z knyhr respectively. What is the average speed for the whole journey? (x y z)/(3(x y+y z+z x)) b. (x y z)/((x y+y z+z x)) c. ((x y+y z+z x))/(x y z) d. (3x y z)/((x y+y z+z x))

Prove that |x^2x^2-(y-z)^2y z y^2y^2-(z-x)^2z x z^2z^2-(x-y)^2x y|=(x-y)(y-z)(z-x)(x+y+z)(x^2+y^2+z^2)dot

If a^(x)=(x+y+z)^(y), a^(y)=(x+y+z)^(z), a^(z)=(x+y+z)^(x) , then :

Prove that Det [[x + y + 2z, x, y], [z, y + z + 2x, y], [z, x, z + x + 2y]] = 2 (x + y + z) ^ 3

Prove that |[x+y, y+z, z+x] , [z+x, x+y, y+z] , [y+z, z+x, x+y]|=2|[x,y,z] , [z,x,y] , [y,z,x]|

A(x, y, z)=" min. "(x+y, y+z, z+x) B(x, y, z)="max "(x-y, y-z, z-x) C(x, y, z)=" max"(A(x, y, z), B(x, y, z)) D(x, y, z)=" min "(A(x, y, z), B(x, y, z)) What is the value of A(2, 3, 4)+B(2, 3, 4) ?

(x + y) ^ (3) + (y + z) ^ (3) + (z + x) ^ (3) -3 (x + y) (y + z) (z + x)