Functional magnetic resonance imaging (fMRI) enables measuring human brain activity, in vivo. Yet, the fMRI hemodynamic response unfolds over very slow timescales (<0.1-1 Hz), orders of magnitude slower than millisecond timescales of neural spiking. It is unclear, therefore, if slow dynamics as measured with fMRI are relevant for cognitive function. We investigated this question with a novel application of Gaussian Process Factor Analysis (GPFA) and machine learning to fMRI data. We analyzed slowly sampled (1.4 Hz) fMRI data from 1000 healthy human participants (Human Connectome Project database), and applied GPFA to reduce dimensionality and extract smooth latent dynamics. GPFA dimensions with slow (<1 Hz) characteristic timescales identified, with high accuracy (>95%), the specific task that each subject was performing inside the fMRI scanner. Moreover, functional connectivity between slow GPFA latents accurately predicted inter-individual differences in behavioral scores across a range of cognitive tasks. Finally, infra-slow (<0.1 Hz) latent dynamics predicted CDR (Clinical Dementia Rating) scores of individual patients, and identified patients with mild cognitive impairment (MCI) who would progress to develop Alzheimer’s dementia (AD). Slow and infra-slow brain dynamics may be relevant for understanding the neural basis of cognitive function, in health and disease.