Let w,x,y and z be four natural numbers such that their sum is 8m+10 where m is natural number. Given m, the following is necessarily true: a) maximum possible value of w^2+x^2+y^2+z^2 is 16m^2+40m+26. Why?

Question

Answer 1

The value of $$w^2+x^2+y^2+z^2$$ is maximum when w = x = y = z
w + x + y + z = 8m + 10 => w = x = y = z = 2m +2.5
$$w^2+x^2+y^2+z^2$$ = $$4(2m+2.5)^2$$ = $$4(4m^2+10m+6.25)$$ = $$16m^2+40m+25$$
Hence, the maximum value is $$16m^2+40m+25$$ and not $$16m^2+40m+26$$

Answer 2

The value of $$w^2+x^2+y^2+z^2$$ is maximum when w = x = y = z
w + x + y + z = 8m + 10 => w = x = y = z = 2m +2.5
$$w^2+x^2+y^2+z^2$$ = $$4(2m+2.5)^2$$ = $$4(4m^2+10m+6.25)$$ = $$16m^2+40m+25$$
Hence, the maximum value is $$16m^2+40m+25$$ and not $$16m^2+40m+26$$

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Let w,x,y and z be four natural numbers such that their sum is 8m+10 where m is natural number. Given m, the following is necessarily true: a) maximum possible value of w^2+x^2+y^2+z^2 is 16m^2+40m+26. Why?