1. ## continuous bounded functions

Let f:[a,b]---->R be Riemann integrable. Define F:[a,b]---->R
by

F(x)= $\int_a^x f(t) dt$
iii) Prove that there exists c in[a,b] such that F(c)=sup{F(x): x in [a,b]}
By boundedness principle a continuous functions attains a maximum and minimum so f is bounded.

iv) Now suppose that f is continuous and that the point c from (iii) satisfies c in (a,b) what can you conclude about f(c)?

I know from lecture note, If f is continuous at c in[a, b] then F is differentiable at c and F'(c) = f(c).
from iii) f(c)=maximum of function so the derivative at maximum value is f'(c)=0 I have been asked for f(c) so can you help please? thanks

2. Originally Posted by charikaar
Let f:[a,b]---->R be Riemann integrable. Define F:[a,b]---->R
by

F(x)= $\int_a^x f(t) dt$
iii) Prove that there exists c in[a,b] such that F(c)=sup{F(x): x in [a,b]}
By boundedness principle a continuous functions attains a maximum and minimum so f is bounded.

iv) Now suppose that f is continuous and that the point c from (iii) satisfies c in (a,b) what can you conclude about f(c)?

I know from lecture note, If f is continuous at c in[a, b] then F is differentiable at c and F'(c) = f(c).
from iii) f(c)=maximum of function so the derivative at maximum value is f'(c)=0 I have been asked for f(c) so can you help please? thanks
Well I believe that F is a continuous function since:

l $\int_a^x f(t) dt$ - $\int_a^c f(t) dt$l <= l $\int_c^x f(t) dt$l <= $\int_a^x lf(t)l dt$<=lx-cl*sup{f(x): x in [a,b]}

but f is bounded right? So sup{f(x): x in [a,b]} = K. Thus for all e>0, set delta = e/K

Thus, iii) follows from the extreme value theorem.

iv) Since c is in (a,b) we have F'(c) = 0

But F'(x) = f(x)

Thus, F'(c) = f(c) = 0

Note: Done while tired and in haste