The product rule states that

Thus, the correct derivative of the function

is:

Now you say

and solve for

, which gives you extreme points at

.

To find the points of inflection or to determine concavity, differentiate once again:

. Your only inflection point is at

.

To verify whether

is a minimum or maximum, check the concavity at those points. So plug in

into

to see if it is positive or negative. If it is positive, then it is concave up (and thus a min) and negative means concave down (and thus a max). Do the same for

.

It helps to graph the function to verify.