$(12$ points$)$ The $1-$D temperature profile within a solid is found to be:

$T=C_1{x}^2+C_{2}lnx+C_3.$

a$.$ Take the first derivative and apply Fourier's law to evaluate the heat flux in the $x-$direction at $x=10,$ in terms of the unknown variables. The thermal conductivity is $k.$

b$.$ If the units of $x$ are $m$ and temperature units are $K,$ what are the units of $C_1?$ What are the units of $C_2?$ Units must balance!

c$.$ Determine the x$-$location of the maximum $($or minimum$)$ temperature by evaluating the first derivative. The constants $C_1<mn>0</mn>$ and $C_2>0,$ and $x>0.$

d$.$ Determine if the temperature is a maximum or minimum at the location in part $(3)$ by checking the second derivative.