Catch-up growth, referring to infants with low birth weight reaching or exceeding normal body weight later in life, is negatively correlated to adult health outcome and life span. Life history theories have suggested that there exist trade-offs between early development and later health maintenance, but detailed mechanisms and the currency of the trade-off are unclear. In this paper, we present a general theoretical model for quantitatively elucidating the trade-off between growth rate and health maintenance in mammals from an energetic viewpoint. Based on the fundamental principles of energy conservation and organisms' energy budgets, our model analyzes the allocation of metabolic energy to growth and health maintenance in different sets of prenatal and postnatal environments. Our model also implies a relationship between growth rate and the general process of aging. Life-span predictions are supported by quantitative and qualitative empirical observations and offer theoretical frameworks for future experimental designs and data analyses.