**a.**$M(T)= $max $u(x,t)$, where $0\leq x\leq l, 0\leq t\leq T]$

We claim that M(T) is an non decreasing function of T.

Let $T_1 \leq T_2$ and the rectangle $R_1 = {0\leq x\leq l,0\leq t \leq T_1} $ and $R_2 = {0\leq x\leq l,0\leq t \leq T_2} $.

The bottom and lateral sides of $R_1$ contained in $R_2$. By the uniqueness property, the diffusion equation of u on $R_2$ is an extension of u of $R_1$ and thus$M(T1)\leq M(T2)$

**b.**$m(T)=$ min $u(x,t)$, where $0\leq x\leq l, 0\leq t\leq T]$

We claim that m(T) is an non increasing function of T.

Let $T_1 \leq T_2$ and the rectangle $R_1 = {0\leq x\leq l,0\leq t \leq T_1} $ and $R_2 = {0\leq x\leq l,0\leq t \leq T_2} $.

The bottom and lateral sides of $R_1$ contained in $R_2$. By the uniqueness property, the diffusion equation of u on $R_2$ is an extension of u of $R_1$ and thus $m(T2)\leq m(T1)$