08wHW2 - f ( x ) = 0 for x [ a, b ]. 3. Let f, g : [ a, b ]...

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MAT 125B Homework 2 M.Fukuda Please submit your answers at the discussion session on January 29 Tuesday. You can use theorems in the lecuture unless otherwise stated. When you do so write those statements clearly instead of quoting them by numbers. 1. Let f : [ a, c ] R and b [ a, c ]. Show that i b a f ( x ) dx = i c a f ( x ) dx + i b c f ( x ) dx. You don’t have to prove this result from scratch. 2. Let f : [ a, b ] R and α, β R with α n = β . Suppose f is continuous on [ a, b ] and α i c a f ( x ) dx + β i b c f ( x ) dx = 0 for all c ( a, b ). Then, show that
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Unformatted text preview: f ( x ) = 0 for x [ a, b ]. 3. Let f, g : [ a, b ] R . Suppose that f is dierentiable and non-decreasing on [ a, b ], and f is continuous on [ a, b ], and that g is dierentiable on [ a, b ] and g is integrable on [ a, b ], and g is strictly positive on ( a, b ) but g ( a ) = g ( b ) = 0. Then, show that f is constant on [ a, b ] if and only if i b a f ( x ) g ( x ) dx = 0 . 1...
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This homework help was uploaded on 04/08/2008 for the course MATH 125B taught by Professor Fukuda during the Winter '07 term at UC Davis.

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