A student asserts: “For any continuous even function g defined on [−L,L] with L>0, the average value of g on [−L,L] equals L1∫0Lg(x)dx.” Which of the following best evaluates the student’s assertion?
a) The assertion is invalid because the average value formula always divides by 2L, even for an even function.
b) The assertion is valid because for an even function ∫−LLg(x)dx=2∫0Lg(x)dx, so 2L1∫−LLg(x)dx=L1∫0Lg(x)dx.
c) The assertion is invalid because symmetry forces ∫−LLg(x)dx=0, so the average would have to be 0.
d) The assertion is valid only if g(0)=0; otherwise the lack of symmetry about the y-axis changes the average value.