For positive real numbers \(x,y,\) and \(z,\) find the minimum value of \(\frac{x^3 + 5y^3 + 25z^3}{xyz}.\)
x=1;y=1;z=1; a=(x^3 + 5*y^3 + 25*z^3) / (x*y*z);printa,y;y++;if(y<100, goto3, discard=0; The minimum value is when: x =2 or x = 3, y=1, z=1: (x^3 + 5*y^3 + 25*z^3) / (x*y*z) = 19
If we set y = x/51/3 and z = x/251/3 then (x3 + 5y3 + 25z3)/xyz = 15 irrespective of the actual (positive) values of x, y and z.