If $E=at+bt^2$ what is the temperature inversion ?

Hint:We know that thermal expansion in the process or the nature by which the physical properties of the given substance varies due to the rapid change in their temperature. We also know that temperature is the measure of heat energy and thus can do some work, which in turn is seen as the change in the physical property.

Formula used:
$Temp\;inversion=\dfrac{dE}{dt}=0$

Complete step-by-step solution:
We know that heat is a form of energy which can be transferred from one system to another. We also know that temperature is the measure of heat energy possessed by the system. However, the concept of hot and cold is relative in nature.
Inversion temperature is the deviation of the atmosphere's property with increase in altitude. Generally, it is observed that air temperature decreases with increase in the altitude, for instance consider a hill top and the beach area, since beach areas are along the sea level they are hotter as compared to that of the hill tops which are way above the sea level.
This is also given as the Joule Thomson effect of inversion temperature, it can be defined as $Temp\;inversion=\dfrac{dE}{dt}=0$. Given that $E=at+bt^2$, where $t$ is temperature
Then from as the Joule Thomson effect of inversion temperature, we have
$\implies Temp\;inversion=\dfrac{d(at+bt^2)}{dt}$
$\implies Temp\;inversion=a+2bt=0$
$\therefore t=\dfrac{-a}{2bt}$
Thus, the temperature inversion of the given equation is temperature $\dfrac{-a}{2bt}$

Note:The study of heat energy or temperature and its constituents is called thermodynamics. Also, heat energy generally flows from the hot body to a cold body in nature; the reverse may require extra effort. The inversion temperature depends on the nature of the gas molecules present during the process.