Suppose x1,...,xn are real numbers. Let x = (x1 +···+ xn)/n. Let c be a real number.

(a) Show that n i=1 (xi − x) = 0.

(b) Show that n i=1 (xi − c)2 = n i=1 (xi − x)2



+ n(x − c)2.

Hint: (xi −

c) = (xi − x) + (x − c).

(c) Show thatn i=1 (xi −c)2, as a function of

c, has a unique minimum at c = x.

(d) Show that n i=1 xi 2 = n i=1 (xi − x)2



+ nx2.