Assigned Exercise IX.1. IX.1. (a) Suppose that f : [a, b] → R is continuous. Define...

Assigned Exercise IX.1. IX.1. (a) Suppose that f : [a, b] → R is continuous. 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 − a integral from a to b of (f(x) − A)² dx = B − A ² . Conclude that A² ≤ B. (b) Assume the Cauchy–Schwarz Inequality for Integrals of Exercise 6.3 #2, which we state here for continuous functions f : [a, b] → R and g : [a, b] → R: integral from a to b of (fg)² ≤ integral from a to b (f ² ) integral from a to b (g ² ) . How does this Cauchy–Schwarz inequality imply the inequality A² ≤ B of part (a)?

Expert Solution

(a)

f : [a,b] |R is continuous.

and thus,

Now, for all x in [a,b], (f(x) - A)² 0 and thus,

which implies that A² B

(b) The Cauchy-Schwartz Inequality for integrals states that:
Suppose F,G : [a,b] |R be continuous . Then,

Now, let F = f and G=1. Applying the Cauchy-Schwartz Inequality on [a,b] yields:

and thus,

( A(b-a) )² B(b-a)(b-a) = B(b-a)² and thus A² B.

Colby Messinger answered 2 years ago

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