Small proof on monotonic functions
I actually made this question up while studying some chemistry. The problem is easy to visualize, but I'm trying to formalize to help myself think more rigorously. To be precise I sort of thought about how you could prove that a reduction in vapor pressure causes a depression freezing point in an ideal solution.
Suppose that [itex]f(x)[/itex] and [itex]g(x)[/itex] are both real-valued differentiable functions defined for all x.
It is known that:
[itex]f'(x)>0[/itex] for all x
[itex]g'(x)>0[/itex] for all x
There exists exactly one value of [itex]c[/itex] such that [itex]f(c) = g(c) [/itex]
There exists a [itex]d[/itex] such that [itex]f(d) = g(d)-5 [/itex]
Prove that [itex]d<c [/itex]
I will be very thankful if someone could help me out. Again this is a problem I made up from my studies in chemistry. Not a homework problem.