I am trying to use Cramer Rao Theorem to estimate $\theta$ when $\{X_i\}_1^n$'s are random samples with probability density function \begin{equation*} f_X(x) = \begin{cases} 3x^2/\theta^3 \text{, $x \in [0,\theta]$} \\ 0 \text{ , otherwise} \end{cases} \end{equation*}
I realized that I have some fatal misundertanding of Cramer Rao Theorem when I tried to find the $CRLB$. This is what I've done:
Likelihood Function: $L(\{x_i\}_1^n,\theta)=\prod_i f_{X_i}(x_i)= \frac{3^n}{\theta^{3n}}\prod_i x_i^2$
Log-Likelihood Function: $l(\{x_i\}_1^n,\theta)=\sum_i \log(f_{X_i}(x_i))= n\cdot \log(3)-3n\cdot\log(\theta)+\sum_i2\cdot \log(x_i)$
Then $\partial_\theta l =\frac{-3n}{\theta}$, and $\partial_\theta^2 l = \frac{3n}{\theta^2} $.
Then, according to the statement of the theorem I will have:
$$Var(\hat{\theta})\geq \frac{1}{n\mathbb{E}[(\partial_\theta l)^2]}=\frac{-1}{n\mathbb{E}[\partial_\theta^2 l]}$$
But, the two terms at the right side are not equal:
$$\frac{1}{n\cdot (\frac{-3n}{\theta})^2}\neq \frac{-1}{n\cdot \frac{3n}{\theta^2}}$$
(One of them is negative!)
Where am I wrong, I would be glad for any help.