minimal suff stats, MLE question (struggling)

Question.

Let be a random sample from the density function

where both and are positive unknown parameters.

i. Find a minimum sufficient statistic for .

ii. Find the maximum likelihood estimator for .

iii. Find the maximum likelihood estimator for .

My attempt at the answer.

i. Joint density for a sample :

Now I need to look at the ratio of two densities,

which will be contant as a function of if and only if , and

or and .

Therefore, is a minimal sufficient statistic for .

ii and iii. Finding MLE for

Use the likelihood function derived in (i)

MLE for .

I note that the likelihood function , if is an independent variable and all other variables held constant, , is increasing and does not attain local maximum. Using the condition , I claim that a attains maximum when it equals the minimum order statistic of the sample.

Therefore, can I say that MLE for is ?

MLE for .

I will maximise the log-likelihood function as follows.

First derivative=0

.

MLE

Second order derivative is negative: therefore this is maximum.