Proof f(x)=0 when integral from a->b equals zero

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Homework Help Overview

The discussion revolves around proving that a continuous non-negative function \( f \) must equal zero if the integral of \( f \) from \( a \) to \( b \) is zero. Participants are exploring the implications of the integral's value and the properties of continuous functions.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • One participant mentions using the Mean Value Theorem to identify a point where \( f(c) = 0 \) but struggles to formalize this argument. Another questions whether the definition of the integral can be leveraged to show that a non-zero function would imply a non-zero area under the curve. There is also a discussion about the suitability of using Darboux or Riemann integrals, with suggestions to construct partitions to analyze the function's behavior.

Discussion Status

The conversation is ongoing, with participants sharing various approaches and questioning the applicability of different integral definitions. Some guidance has been offered regarding the use of partitions and the properties of integrals, but no consensus has been reached on a definitive method or argument.

Contextual Notes

Participants are navigating the constraints of the problem, including the requirement that \( f \) is continuous and non-negative, as well as the implications of the integral being zero. There is an acknowledgment of the potential complexity in proving the statement without additional assumptions or information.

sleventh
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Hello all,
suppose f is a continuous non negative function if int(f(x),x=a..b)=0 show that f(x)=0

what i have done is used mean value theorem to show some point c is such that f(c)=0. from here though i can only think of a verbal argument (since f is non negative) to explain why f(x)=0.

i am wondering if there us an obvious use of the fundamental theorems here that I am not seeing, or just some simple method.

thank you for any help
 
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Straight from the definition of an integral. If there's a point where f(x) is non-zero, can you prove there must exist some small amount of area underneath the graph around that point?
 
you mean darboux or reimann integral? i feel like this might be easier to solve then using sums. perhaps it can be shown the indeterminate integral is a constant function?
 
I think Darboux is a little bit more comfortable here. Just construct a partition such that in at leas one of the intervals f is greater than some fixed \delta > 0.
 

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