With this exact solution, the steady state of T can be determined. This occurs when T approaches a constant value and stays there. When the body approaches the ambient temperature the graph approaches a horizontal asymptote. This is because the rate of change of temperature is approaching zero when the difference between the current temperature and the ambient temperature is approaching zero as well. It starts at the normal temperature of the body and then as the body starts to cool the graph rapidly decreases.
T is the current temperature and is always decreasing to approach the ambient temperature.
Until T equals the ambient temperature, the constant, k, will make T' negative. Because T' represents the slope of T, the graph of T is decreasing. This equation shows that T is always concave up because the second derivative is always positive. The current temperature can never be less than the ambient temperature so T-A will always be positive. It can be verified that the exact solution is correct by plugging it back into the differential equation and initial conditions.