Question

If f is a continuous function on the closed interval [a, b], which of the following must be true?
(A) There is a number c in the open interval (a, b) such that f(c) = 0
(B) There is a number c in the open interval (a, b) such that f(a) < f(c) < f(b).
(C) There is a number c in the closed interval [a, b] such that f(c) ≥ f(x) for all x in [a, b]
(D) There is a number c in the open interval (a, b) such that f'(c) = 0
(E) There is a number c in the open interval (a, b) such that f'(c) = ( f(b) - f(a) ) / (b - a)

Answer 1

would it be C ?

Answer 2

Hello?

Answer 3

Yes, only C is correct

Answer 4

to me it can be E

Answer 5

okay thanks!
I choose E before and got it wrong :P

Answer 6

no, E is not true. Consider f(x) = |x| for [-1,1]

Answer 7

I can not draw
if f(x)=x in [2,4] ,then what ?

Answer 8

Here c is not true.

Answer 9

there should be additional information like f(x) is derivable in open interval (a,b)
or it can be f(x) is strictly increasing or strictly decreasing.