Maximum Likelihood Estimate and Second derivative test? We're doing MLE's in my prob stats class currently, and my professor insists that after we get the derivative of the Ln of L we still need to take the second derivative to check if it's the MLE. My book however says that the derivative of the Ln of L is always the MLE. He refuses to let us skip this step without a Theorem/proof for this, but I can't find one. Can anyone point me towards one, or correct me if our book is wrong?
 A: The MLE is simply the global maximum of a  function (the likelihood , as a function of the parameter ): $L(\theta)$
You are supposed to know, not from Statistics but from Calculus, that, in general, the global maximum of a function cannot simply be found by deriving the function and equating it to zero. That only gives as a  critical point (perhaps several). It can well happen that


*

*the function is not derivable in some points of the domain

*the (some) critical point is not a maximum

*the global maximum occurs on the boundary of the domain


The estimation of an uniform variable on $[0,\theta]$ is an example of the last situation.
Granted, if you know that


*

*$L(\theta)$ is derivable in all its domain

*there is a single critical point

*the domain of the parameter is the whole real line

*$L(\theta)$ tends to zero at $\pm \infty$ 


then you know that the critical point is the MLE. If you know the first 3 conditions but not the fourth, then you should compute the second derivative.
If you only know $1.$ and $2.$ then you should also check the boundaries of the domain.
