Hi everyone. I’m struggling to completely solve the following exercise:
• Study the function , finding its solution/s, critical point/s and inflexion point/s.
Firstly, the unique solution is
Secondly, to find the critical points we have to derivate the function and make it equal to 0. Therefore:
Then the critical points are:
As we know, we have to do a second derivative and calculate it in function of the critical points’ value, in order to find the inflexion, maximum and minimum point/s. Therefore:
A theorem tells us this result is inconclusive
Then we have a maximum
As the function is odd we can directly claim than we have a minimum at the critical point :
Therefore we have a minimum
Once I arrive here, my struggle starts. I know that there is a method using Taylor to analyse the case:
It allows us to determine if there’s such an inflexion point and, what’s more, find out how can be drawn.
I have looked for this method in many books: Calculus of Marsden and Weinstein, Stewart...
But I find it nowhere.
Could someone explain me how to use the method and finish this problem?
if it is not possible to explain here in a detailed way, could you provide me a reference where I could understand it deeply?
Thank you very much