Consider the function
First note that is differentiable (why)
Now use the mean value theorem on
Can you finish from here
Here's the question:
a.) Prove that if f is integrable on [a,b] and m <= f(x) <= M for all [a,b], then:
for some number "k" with m <= k <= M
b.) Prove that if f is continuous on [a,b], then
for some q in [a,b]
So for the work I set up that U(f,P) - L(f,P) < E, which implies the usualy equasion of the Sum of (Mi-mi)(Change in ti) < E. But I'm unsure where I should start throwing in this "k" variable? Thanks
Would F'(x) = f(x), then? So F'c = f(c). In this case does this f(c) become our k? I'm still having some trouble trying to figure out why exactly F'(c) does stay between m and M. Could I say f(c) (or k) would have to be between m and M due to previous theorems regarding continuity (if f(a) < c < f(b), there exists some x with f(x) = c? and that we do obviously have some bounds?
Thanks EmptySet! I actually figured out part a. right before you posted, but I did the exact same steps so I'm glad. An add-on to this is a question that asks "Show by an example that continuity is essential". I'm unsure of what he is asking, because I know that we can have discontinuous functions that still can be integrated. Continuity is sufficient but not necessary for integrability right?