We present an analysis of the growth of black holes through accretion and bulges through star formation in 33 galaxies at the centers of cooling flows. Most of these systems show evidence of cavities in the intracluster medium (ICM) inflated by radio jets emanating from their active galactic nuclei (AGN). We present a new and extensive analysis of X-ray cavities in these systems. We find that AGN are energetically able to balance radiative losses (cooling) from the ICM in more than half of our sample. Using a subsample of 17 systems, we examine the relationship between cooling and star formation. We find that the star formation rates are approaching or are comparable to X-ray and far UV limits on the rates of gas condensation onto the central galaxy. The remaining radiative losses could be offset by AGN feedback. The vast gulf between radiative losses and the sink of cooling material, which has been the primary objection to cooling flows, has narrowed and, in some cases, is no longer a serious issue. Using the cavity (jet) powers, we place strong lower limits on the rate of growth of supermassive black holes in central galaxies, and we find that they are growing at an average rate of ~ 0.1 solar masses per year, with some systems growing as quickly as ~ 1 solar mass per year. We find a trend between bulge growth (star formation) and black hole growth that is approximately in accordance with the slope of the local (Magorrian) relation between black hole and bulge mass. However, the large scatter in the trend suggests that bulges and black holes do not always grow in lock step. (Abridged)
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