Suppose f (x) is a continuous function on [0;1], and 0 <= f(x) <= 1 for all x any [0;1].
(a)Show that f (x)= 1 - x for some number x.
(b)Prove the more general statement: Suppose g is continuous on [0,1] and g(0)= 1, g(1)= 0,then f(x)= g(x) for some number x.
The Attempt at a Solution
Neither makes sense to me. For the first one, isn't f(x) = x completely valid? Or a compressed sin function? Are they asking to show that 1-x is also valid, or that f(x) must be 1 - x no matter what?
For the second one, isn't a compressed cos function valid? It will be continuous on [0,1], g(0) = 1 and g(1) = 0. So why does g(x) = f(x) = 1-x only?
The question is copied word for word so tell me if I missed or misunderstood anything.