Namaste!

Let where

Find the interval where is concave up, and where is concave down. Finally, list any points of inflection (if there are any).

I began by taking the first derivative of :

(not 100% sure this is correct)

Then I found the second derivative of and got:

To find the critical points I (attempt to) set the second derivative to zero to find where the graph of is concave up or concave down.

Broken into steps:

I try to think of where the graph of

Finally:

So this is the critical point, and possibly a point of inflection.

I test points on a number line on either side of this to find where the second derivative is positive or negative to find where the original function is concave up, or concave down.

A hint was given to use a double angle formula, however I am not very familiar with how to implement these.

However, all of my answers so far have been incorrect. I don't want the solution, I just want to know where I went wrong through this process. Any help would be greatly appreciated. Thanks in advance.