Many biological variables depend on body mass, with a functional relationship of the form Y = Y₀ M^b, where M represents body mass, b is a multiple of 1/4, and Y₀ is a constant. For example, when Y represents metabolic rate, b = 3/4. When Y represents heartbeat, b = -1/4. When Y represents life span, b = 1/4.

(a) Determine which of metabolic rate, heartbeat, and life span are increasing or decreasing functions of mass. Also determine which have graphs that are concave upward and which have graphs that are concave downward.
(b) Verify that all functions of the form given above satisfy the equation

This means that the rate of change of Y is proportional to Y and inversely proportional to body mass.

XP dM || b M Y.