Likelihood and maximum likelihood

Question

what is the likelihood, log-likelihood and MLE of;

$$θ(θ+1)x^{θ−1}(1−x)$$

any help greatly appreciated

Answer 1

Just to tidy up some of your work (too long for a comment) $$ \begin{align} \text{Likelihood} &=& \theta^n(\theta+1)^n\prod x_i^{\theta-1}(1-x_1)\\ \text{Log-Likelihood} &=& n\log \theta + n\log (\theta+1) + \\ &+&(\theta-1)\sum x_i + \sum\ln(1-x_1) \end{align} $$ To answer your question in the comments, you cannot sum up the observations the same way you can sum with the estimator, this is because, we are assuming there is one estimator to describe the observations.

Now can you determine the MLE (bear in mind it will be the same if you utilize either of the above)