I found this question to be quite intimidating:
Suppose a function f has the following properties:
a) Show that:
b) Show that for all values of x by first showing that for all x.
c) Use the definition of the derivative to prove that for all x.
I'd assume f would be a constant function. It seems to have similar properties as such a function. Anyway, thanks for help!
well since you KNOW f(0) = 1, then:
1 = f(0) = f(a+(-a)) = f(a)f(-a).
this means that f(a) ≠ 0 (or else we would get 1 = 0, right?). this works for ANY a.
note that since f(a)f(-a) = 1, f(a) and f(-a) always have the same sign. so on the interval (-a,a) = (0-a,0+a) f is either entirely positive, or entirely negative, and thus positive (since f(0) = 1 > 0).
(this is where continuity comes in: suppose for some b > 0, we had f(b) < 0. by continuity, at some point in (0,b), say c, we must have f(c) = 0 (the intermediate value theorem). but there IS no such c. a similar argument works for any b < 0).
but if f is positive on any interval (-a,a), it must be positive on all of R.
c) is relatively easy: by definition-
, so....
also: your initial guess that f is constant is wrong: if f was constant, then f'(0) would always be 0 (constant functions are "flat").
one class of (continuous) functions for which f(x+y) = f(x)f(y) are THESE: f(x) = a^{x}, for a > 0. the trouble with this is, what does a^{x} even MEAN for an irrational x?
one way of doing this is by imagining a continuous function a^{x} that "connects the dots" where "the dots" are a^{x} for RATIONAL x. it would be helpful if we could produce a specific "a" that we KNEW existed (besides 1, which is boring). that is what this problem is trying to do: figure out what such a function might be like.
what's coming next is: for what function f(x) = a^{x} (if it exists, and is differentiable) might we have f(x) = f'(x)?
Since we are given that f is differentiable, it is continuous and it is relatively easy to show that (equivalently [/tex]f(x)= r^x[/tex]) are the only continuous functions satisfying f(x+y)= f(x)f(y). There are many NON-continous functions satisfying that but they have to be really discontinuous! In fact, they must be unbounded on any interval.
my point was: how do you define the function f(x) = e^{x} if one doesn't already know what it is? what's so special about "e" that (e^{x})' = e^{x}, but not for other positive real numbers?
but, we have something we've learned about f(x) (just ignore e for a bit, pretend you've never heard about it). it equals its own derivative, AND...this derivative (itself) is always positive. that is: f(x) is a strictly increasing function.
and THAT means: for every f(x) in the range of f, we have an inverse function g:f(x)--->x.
here's the cool thing about inverse functions:
suppose we have for a,b in R: [f(b) - f(a)]/(b - a), and f has an inverse on its entire domain (so f^{-1} is defined for all f(x) in the range of f).
suppose that f(a) = c, and f(b) = d. we can re-write the fraction above as:
(c - d)/[f^{-1}(c) - f^{-1}(d)]
in particular, if b = a+h:
[f(a+h) - f(a)]/h = (c - d)/[f^{-1}(c) - f^{-1}(d)]
suppose that we call d-c, k, so that d = c+k.
then:
[f(a+h) - f(a)]/h = -k/[f^{-1}(c) - f^{-1}(c+k)] = k/[f^{-1}(c+k) - f^{-1}(c)]
if f is continuous, then as h-->0, f(a+h) - f(a) -->0, which means that d-c -->0, that is: c+k - c = k -->0.
so:
if the derivative of f is never 0 for any of the x's we're considering:
we can also write this as:
which will be more useful for us.
what does this mean for the PARTICULAR f(x) we've been discussing in this thread? to simplify the notation, let's use g(y) instead of f^{-1}(y).
it says that:
but, in this case: f'(g(y)) = f(g(y)) = y (since f and g are inverses, and since f'(x) = f(x)). that is:
g'(y) = 1/y.
note that since f takes x+y to f(x+y) = f(x)f(y)
we have g(f(x)f(y)) = g(f(x+y)) = x + y = g(f(x)) + g(f(y))., so writing f(x) = a, f(y) = b:
g(ab) = g(a) + g(b). holy canoli! we've discovered LOGARITHMS!
also, since f(0) = 1, g(1) = 0.
if you know the fundamental theorem of calculus, you know that:
so for our newly-discovered function g:
that is:
.
now, all along we're thinking f(x) = a^{x} for some positive number a. now we have a way of finding a, it is the real number a such that:
<-of course we have to have some means of approximating this integral, which might be tough, but we CAN do it.