Because f is continuous on [a,b] it has a maximum and minimum there.
Let .
By continuity .
Now we have .
Using the intermediate value theorem finish.
Let f be continuous on the closed and bounded interval [a,b] and let x1, x2, .... , xn be in [a,b]. Show that there must exist a c in [a,b] such that f(c) =
f(x1) + f(x2) + .... + f(xn)
----------------------------------------.
n
I think all I have to show is that:
minimum value of f(x) in [a,b] is less than or equal to the expression above which is less than or equal to the maximum value of f(x) in [a,b]. But how do I show that this is true? Then, I can just use the Intermediate Value Theorem to prove that a f(c) exists.
Thank you for any help (the ------------ is obviously a division sign).