Background: Alzheimer's disease (AD) affects over 55 million people worldwide. Accurate, interpretable detection of normal cognition (NC), mild cognitive impairment (MCI), and AD from routine clinical assessments remains a critical unmet need. Methods: An XGBoost classifier was developed for three-class detection using eight clinical features from the Alzheimer's Disease Neuroimaging Initiative (ADNI): MMSE, CDR Global, CDR Sum of Boxes (CDR-SB), MoCA, FAQ, age, sex, and education. Hyperparameters were optimised using Optuna (50 trials); class imbalance was addressed with SMOTE. Performance was evaluated by macro AUC-ROC with 1,000-iteration bootstrap 95% confidence intervals, macro F1, balanced accuracy, and Cohen's kappa. SHAP values provided feature-level explainability. Results: The dataset comprised 1,641 baseline subjects (608 NC, 767 MCI, 266 AD). On five-fold cross-validation, mean macro AUC was 0.983 (SD 0.007), accuracy 0.944 (SD 0.006), and macro F1 0.929 (SD 0.008). On the held-out test set (n = 247), macro AUC was 0.982 (95% CI: 0.965--0.995), accuracy 0.943, balanced accuracy 0.932, macro F1 0.927, and Cohen's kappa 0.909. SHAP analysis identified CDR Global as the dominant predictor for NC and MCI, while CDR-SB and MMSE together drove AD classification. Conclusion: An explainable machine learning model trained on routine clinical assessments achieves near-perfect three-class Alzheimer's detection. SHAP analysis reveals clinically plausible, class-specific feature importance patterns supporting clinical validity. Future work will extend this framework with speech biomarkers for multimodal detection.