Functional relationship inequality

T

Thanasis

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I have find the monotony but I do not know how to to solve the inequality. I know that I have to use the monotony but I am not sure how. I have show what I do for the first part.11220
 
You know f' (x), now take the derivative of the inequality expression and then verify that the inequality sign is correct
 
Without an expression so we can calculate what f(x) is I don't see how to do part ii.

Is this the whole question?

-Dan
 
I see an interesting clue that may lead to a solution of (ii): You can find f(ln(1/2)), by plugging ln(1/2) in for x in the defining equation, and solving for y by inspection. Since you know f is monotonic, it is invertible, and any value you find for y is unique (and in fact you can show that directly). This puts the equation you are to solve in an interesting form that is again related to monotonicity. I haven't taken it beyond that yet.
 
I hadn't thought about the f(ln(1/2)) part. I was thinking more about solving the cubic for [math]f(2^x + x)[/math].

-Dan
 
I think I found the answer by finding f(ln1/2). I will post it soon. I hope i am right?
 
Finally after 2 days I found the solution. Honestly I knew that f(ln1/2) had to be one but for some silly reason I could not prove that. Here is the solution1122411225
 
Looks good. I think I stopped where I did because I wrote a sign wrong, so it didn't look like it had a nice solution.

Interesting problem!
 
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