So we have .
Let . Then is a mean value of f.
f is continuous on [a,b], then f has the Darboux property. So, exists such as .
Don't really know how to prove this at all, I'm much appreciative of any help.
Let f : [a,b] ⇒ ℝ be a continuous function. Show that there exists a w in [a,b] such that
a
∫ f(t)dt = f(w)(b-a)
b
Here is the hint that i need to use to solve this problem:
f attains a smallest value A and a largest value B. Show that:
-----------a
A ≤ 1/(b-a)∫ f(t)dt ≤ B
-----------b
Thank you again for the help!