Expert: Paul Klarreich - 12/4/2009

Question
I am not sure how to start this one.
Suppose that f and g are Riemann integrable on [a,b] and are such that the integral of f from a to b is less than or equal to the integral of g from a to b.
a. Is it true that f(x) is less then or equal to g(x) for all x in [a,b]? Explain.
b. Is it true that f(c) is less then or equal to g(c) for some c in [a,b]? Explain.

I think is part a is true then part b is also true.

Answer
Questioner: Sam
Country: United States
Category: Advanced Math
Private: No
Subject: Riemann Integrable
Question: I am not sure how to start this one.
Suppose that f and g are Riemann integrable on [a,b] and are such that the integral of f from a to b is less than or equal to the integral of g from a to b.
a. Is it true that f(x) is less then or equal to g(x) for all x in [a,b]? Explain.
b. Is it true that f(c) is less then or equal to g(c) for some c in [a,b]? Explain.

I think is part a is true then part b is also true
..............................................

You are being asked about this theorem (which, being a known theorem, certainly is true):

If  f(x) <= g(x) on [a,b], then

{b      {b
| f  <= | g
{a      {a

So part a. of your question is NOT TRUE.  Obvious counter-example:

Try  f(x) = sin x on [0,pi].  The integral is 2.
Try  g(x) = 0.7   on [0,pi].  The integral is 0.7 pi = 2.2 or so.

So g(x) has a bigger integral.  But is there any point where sin(x) > 0.7?  You answer that.

But part b. of your question is TRUE, because it is exactly the equivalent of the THEOREM above.

You wrote:

Is it true that f(c) is less then or equal to g(c) for some c in [a,b]? Explain.

Well, if  f(c) is NEVER <= g(c) on [a,b], then f(c) is ALWAYS > f(c), and then we would have:

{b      {b
| f  >  | g
{a      {a