Parameter Estimate Example: MoM, MLE (1)

Example) I have no idea where I found this example, Sorry!

Let Y1, Y2, …, Yn denotes random sample. f(y|$\theta$)= $(\theta+1)y^\theta$, 0$<$y$<$1, -1$<\theta$, and f(y|$\theta$)=0 otherwise. 
a) $\hat{\theta}_{MoM}$ 
b) $\hat{\theta}_{MLE}$?

$\triangleright$ Solution (a)
$E(\theta)=\int_{0}^{1}y(\theta +1)y^\theta dy=\int_{0}^{1}(\theta +1)y^{\theta +1}dy =\frac{\theta +1}{\theta +2}\Rightarrow \bar{y}=\frac{\theta +1}{\theta +2}$

$\therefore \theta_{MoM}=\frac{2\bar{y}-1}{1-\bar{Y}}$ 

$\triangleright$ Solution (b)
$L(\theta|y)= \prod_{i}^{n}=f(y_{i}|\theta)=\prod_{i}^{n}(\theta+1)y_{i}^\theta=(\theta+1)^n\cdot(\prod y_{i})^\theta$ 

Now we take a log to find maximum likelihood. 
$l(\theta)=nlog(\theta+1)+ \theta\sum logy_{i}$ 
$l'(\theta)=\frac{n}{\theta+1}+\sum y_{i} = 0$  (set to 0 to find a maximum)

$\therefore \hat{\theta}_{MLE}=\frac{-n}{\sum log y_{i}}-1$