Please help solve this, I can't come up with a solution. The question is to find f? Given #f:(0,+oo)->RR# with #f(x/e)<=lnx<=f(x)-1 , x in (0,+oo)#
The answer is f(x) = lnx +1 but how do i prove it?
The answer is f(x) = lnx +1 but how do i prove it?
2 Answers
Explanation:
We split the inequality into 2 parts:
Let's look at (1):
We rearrange to get
Let's look at (2):
We assume
From the 2 results,
Assume a form then use the bounds.
Explanation:
Based on the fact that we see that f(x) bounds ln(x), we might assume that the function is a form of ln(x). Let's assume a general form:
Plugging in the conditions, this means
We can subtract
Flipping,
If we want this to be true for all x, we see that the upper bound is a constant and
So we have only the solution with