I have been asked to justify if the following statement is true or false.
'If the integral of f(x) >= 0 on [a,b], then f(x) is >= 0 on [a,b].
Would (8-x^2) over the domain [-4,4] be a counter example to this? thus justifying that the statement is false.
I beleive so as f(x) is <= 0 for some values but the area over that domain is positive. Can anyone tell me if iam on the right path?
Your counter example is correct, since
however
I think you were thinking of the other direction of the question.Definitely not.
I think it should be obvious that if a function is nonnegative for an entire region, then the area between the function and the axis can never fall below the axis.