take derivative .
Now .
And so .
Thus, and that is the answer.
I have and I must find . I'm asking you to help me finding the inverse function of f, I'll try to do the derivative.
One of my guesses would be that as I have an integral of a function, the inverse operation is the derivative of the integral. In other words I would say that maybe the inverse function of is . But looking back, if I replace this in the upper limit of the integral of , I doubt it will give me the identity. I also know that is . Hmm... can you help me a bit?
Nicely done! I wouldn't think about it... I understand everything except one thing : . Why does it imply that ? Is it because , replacing by we would get it? If yes, then I understood everything, and that's a beautiful answer.
P.D.: Just curious if it's possible to get the general formula . Because, as I said, it's maybe equal to the derivative of . Note that, maybe by a coincidence we have that .
EDIT : Nevermind my P.D. I just realized it's totally false and worthless.
f is strictly increasing (But note that f'(x) is equal to 0 in some separate points, but I think it means that f is strictly increasing and has critical points other than extrema, but I'm not sure.) and continuous. Does that make f one-to-one, or have I to work harder due to the fact that f'(x) can be equal to 0?By the way, I never proved the function is invertible. I just assumed that. That is something you need to prove yourself.