We know that the Fisher information for a probability density function $f_{{\theta}}(\textbf{X})$ with a given true value of the parameter ${\theta}$ and an observable random variable $\textbf{X}$ is given by
\begin{equation} {\small \begin{split} I(\theta) = E\left[\left(\frac{\partial \ln f_{{\theta}}(\textbf{X})}{\partial \theta}\right)^2\right]= \frac{1}{N}\sum_{i=1}^{N}\left(\frac{\partial \ln f_{{\theta}}(\textbf{X}^i)}{\partial \theta}\right)^2 \label{fishinfo} \end{split} } \end{equation} Score is given by $\frac{\partial \ln f_{{\theta}}(\textbf{X})}{\partial \theta}$. At the true value of $\theta$, $\frac{\partial \ln f_{{\theta}}(\textbf{X})}{\partial \theta}=0$, am I rite in saying this?
If this is true how is $I(\theta)$ not zero? I am unable to understand the concept here. Any help will be appreciated.