Question 2. In this problem we give another characterization for the average function value introduced in lecture. Let a < b be constants and let f be an integrable function. Consider the function F:R → R, defined by F(x) = | (f(t) – x)² dt. dF (1) Find the critical point(s) of F (i.e., the point(s) where = 0). da (2) On such critical point(s) does the function F achieve a minimum, maximum, or neither? 1 (3) Based on parts 1 and 2, characterize the average function value :/ f(t) dt in terms of the function b - a F.
Question 2. In this problem we give another characterization for the average function value introduced in lecture. Let a < b be constants and let f be an integrable function. Consider the function F:R → R, defined by F(x) = | (f(t) – x)² dt. dF (1) Find the critical point(s) of F (i.e., the point(s) where = 0). da (2) On such critical point(s) does the function F achieve a minimum, maximum, or neither? 1 (3) Based on parts 1 and 2, characterize the average function value :/ f(t) dt in terms of the function b - a F.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:Question 2. In this problem we give another characterization for the average function value introduced in lecture.
Let a < b be constants and let f be an integrable function. Consider the function
F : R → R, defined by F(x) = | (f (t) – x)² dt.
dF
(1) Find the critical point(s) of F (i.e., the point(s) where
= 0).
dx
(2) On such critical point (s) does the function F achieve a minimum, maximum, or neither?
1
(3) Based on parts 1 and 2, characterize the average function value
| f(t) dt in terms of the function
а
F.
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