Math Help Forum: Integrals and Proofs

Hey there,

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

Consider the function



First note that is differentiable (why)

Now use the mean value theorem on



Can you finish from here

F(x) is differentiable because we have assumed that the function f(t) is integrable?
From there, using the mean value theorem I would see that in the end we'd get





Does this end up meaning F'(c) = k? and thus we have our result?

Remember what the function

represents and you still need to show that


What is equal to?

So the question is why is for some

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?

Originally Posted by mscbuck

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?

hmm. I think that this proof will only work for part b when f is continuous.

The function can only be differentiated by the FTC when is continuous.

So here is a different proof:
Since




Now choose

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?

Originally Posted by mscbuck

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?

Yes that is correct consider the function



and the integral