Assigned Exercise IX.1. IX.1. (a) Suppose that f : [a, b] ? R iscontinuous. Define A := 1/b?a integral of f from a to b, and B :=1/b-a integral of f2 from a to b . Show that 1/b ? aintegral from a to b of (f(x) ? A)2 dx = B ? A2 . Conclude that A2 ? B. (b) Assume theCauchy–Schwarz Inequality for Integrals of Exercise 6.3 #2, whichwe state here for continuous functions f : [a, b] ? R and g : [a,b] ? R: integral from a to b of (fg)2 ? integral from ato b  (f 2 ) integral from a tob  (g 2 ) . How does this Cauchy–Schwarzinequality imply the inequality A2 ? B of part (a)?
