4.Let x , y and z be three natural numbers such that x+y+z=9m+10, where m is a natural number. For any m,which of the following holds true? 1. The minimum possible value of x^2 + y^2 + z^2 is 27m^2 - 60m + 34. 2.The maximum possible value of x^2 + y^2 + z^2 is 27m^2 + 60m + 34 3.The minimum possible value of x^2 + y^2 + z^2 is 27m^2 + 60m + 34 4. The minimum possible value of x^2 + y^2 + z^2 is 27m^2 - 60m + 34
x + y + z = 9m + 10
and x^{2} + y^{2} + z^{2} = (x + y + z)^{2} - 2(xy + yz + zx) is minimumwhen (xy + yz + zx) is maximum.
And ( xy + yz + zx) is maximum when x = y = z = 3m + 10/3.
Substituting values for (x + y + z) as 9m + 10 and x, y and z
x^{2} + y^{2} + z^{2} _{minimum} = 27m^{2} + 60m + 34. Option 3 is correct. |