Suppose that functions f,g : [a,b] -> R are continuous, satisfy f(a) <= g(a)
and f(b) >= g(b). Then there exists a real number c in [a,b] such that f(c) = g(c).
Label the statement as true or false. If it is true, prove it. If not, give an example of why it is false and if possible, correct it to make it true.© BrainMass Inc. brainmass.com March 4, 2021, 6:41 pm ad1c9bdddf
This one can be done as follows: We have f(x) and g(x), x is an element of [a,b]. f and g are continuous and real-valued on [a,b]. We have f(a)<=g(a), f(b)>=g(b).
There are 4 possibilities:
1. f(a)=g(a), f(b)>g(b)
2. f(a)<g(a), f(b)=g(b)
3. f(a)=g(a), f(b)=g(b)
4. f(a)<g(a), ...
Properties of Continuous Functions and the Intermediate Value Theorem are investigated.