Over a certain temperature range, the thermal conductivity k of a metal is not constant but varies as indicated in figure . A lagged rod of the metal has its ends maintained at temperatures $T_{1}and{T}_2(T_2>T_1)$ as shown in figure–  
                             
Which one of the following correctly describes how T₃, the temperature at the mid-point of the rod, compares with T₁ and T₂ ?

The rate of heat flow per unit cross sectional area along the metal rod is given by
  $\frac{Q}{tA}=-k\frac{dT}{d\mathcal{l}}$
where  $\frac{dT}{d\mathcal{l}}$  is temperature gradient along the rod
$\frac{Q}{tA}$  is the same at all points.
$\therefore$   $-k\frac{dT}{d\mathcal{l}}$ = constant                   … (1)
If k is a constant which remain the same over the temperature range, the variation of temperature will be uniform throughout the rod as shown below and temperature T3 will be mean of  $T_{1}and{T}_2$. 

Now k, the thermal conductivity is not constant but varies as indicated in graph of question. 
At  $T_1$ the value of k is the highest thus the temperature gradient  $\frac{dT}{d\mathcal{l}}$  will be the smallest . This indicates that temperature variation will be small initially. As the distance increases, k decreases and thus the temperature gradient increases as it can be readily observed. 
Thus the temperature variation (drop) along the rod increases as distance  $l$  increases from 0 to L. We thus expect the temperature curve convex away from the origin and hence the temperature at mid point of rod $T_3$ will be greater than mean of  $T_{1}and{T}_2$ i.e. $T_3>\frac{{\displaystyle T_1+T_2}}{{\displaystyle 2}}$