a. Show that the point c guaranteed to exist by the Mean Value Theorem for f(x) = x² on [a, b] is the arithmetic mean of a and b; that is, c = (a + b)/2.

b. Show that the point c guaranteed to exist by the Mean Value Theorem for f(x) = 1/x on [a, b], where 0 < a < b, is the geometric mean of a and b; that is, c = √ab.