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Let f, g € R[a,b]. Suppose that g > 0 on [a, b]. 1. Show that inf f - [a,b] g(x) dx < :| f(x)g(x) dx < sup f [a,b] g(x) dx. a

Let f, g € R[a,b]. Suppose that g > 0 on [a, b]. 1. Show that inf f - [a,b] g(x) dx < :| f(x)g(x) dx < sup f [a,b] g(x) dx. a

Advanced Engineering Mathematics
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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Let f, g E R[a, b]. Suppose that g > 0 on [a, b]. 1. Show that inf f · [a,b) | f(x)g(x) dæ < sup f [a,b] g(x) dæ < g(x) dx. a 2. Assume additionally that f E C[a,b]. Show that there is § E (a, b) such that | f(x)g(x) dx = f(£) / g(x) dx. Hint: use the intermediate value property for f.
Transcribed Image Text:Let f, g E R[a, b]. Suppose that g > 0 on [a, b]. 1. Show that inf f · [a,b) | f(x)g(x) dæ < sup f [a,b] g(x) dæ < g(x) dx. a 2. Assume additionally that f E C[a,b]. Show that there is § E (a, b) such that | f(x)g(x) dx = f(£) / g(x) dx. Hint: use the intermediate value property for f.
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