Patients with Alzheimer's disease (AD) often display disrupted resting-state brain functional connectivity compared to healthy individuals, as observed in resting-state fMRI studies. Understanding which brain regions contribute to this disruption is vital for understanding disease progression.We propose the Functional Connectivity Curve Detection model (FunCurvDtx) to identify key brain regions for classifying AD versus normal subjects based on functional connectivity. FunCurvDtx combines functional principal component analysis (fPCA) and group penalized logistic regression to analyze functional connectivity matrices. In FunCurvDtx, a functional connectivity curve (FC curve) is constructed for each seed brain region, with the Euclidean distance representing its domain and functional connectivity serving as the corresponding function value. This approach accounts for spatial correlations among adjacent brain regions within the functional connectivity matrix. By employing fPCA, FunCurvDtx extracts functional component scores (FC scores) from these curves, reducing the infinite-dimensional space of FC curves to a finite set of informative features. With 164 defined brain regions, each associated with approximately 15 FC scores, the model organizes these variables into groups, facilitating effective analysis. To capitalize on inherent group information, FunCurvDtx utilizes group penalties within logistic regression to identify the most salient brain regions associated with the presence of AD. Real data analysis demonstrates the model's superior performance compared to existing methods and provides insights into the effects of the selected brain regions on AD pathology. Overall, FunCurvDtx presents a promising framework for unraveling the intricate interactions among brain regions in the context of AD.