Let f ; g R[a, b]. (a) If t R, show that ba (tf

Question:

Let f ; g ∈ R[a, b].
(a) If t ∈ R, show that ∫ba (tf ± g)2 > 0.
(b) Use (a) to show that 2|∫ba f g| < t ∫ba f2 + (1/t) ∫ba g2 for t > 0.
(c) If ∫ba f2 = 0, show that ∫ba f g = 0.
(d) Now prove that |∫ba f g|2 < (∫ba | f g|)2 < (∫ba f2) ∙ (∫ba g2). This inequality is called the Cauchy-Bunyakovsky-Schwarz Inequality (or simply the Schwarz Inequality).