integral Q: if int[a,b] [f] dx = 0, prove f = 0 for all x in

[dopey9]

How do I show that if:

. . .\(\displaystyle \L \int_a^b\, f(x)\,dx\, =\, 0\)

...then f(x) = 0 for all x in the interval [a, b]?
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Edited by stapel -- Reason for edit: formatting

 

[stapel]

I don't think the statement is true: Consider f(x) = cos(x) over the interval [a, b] = [0, pi].

Eliz.

 

[dopey9]

something i forgot

I don't think the statement is true: Consider f(x) = cos(x) over the interval [a, b] = [0, pi].

Eliz.

i forgot to mention that f:[a,b] -> is a positive continious function i.e. f(x)>= 0 for all x in [a,b]

and then the question i posted sorrry..thank you

 

[pka]

Re: something i forgot

f:[a,b] -> is a positive continious function i.e. f(x)>= 0 for all x in [a,b]

Well that changes everything. If a continuous function non-negative on [a,b] and if f(c)>0 for some c in [a,b] then f is positive of some subinterval [d,e] of [a,b]. Thus, the integral has to be positive.