Check that function $u=x^3+6xt$ satisfies diffusion equation $u_t-u_{xx}=0$ and find

\begin{align*}

&M(T)= \max _{0\le x\le L,\ 0\le t\le T} u(x,t),\\[2pt]

&m(T)= \min _{0\le x\le L,\ 0\le t\le T} u(x,t).

\end{align*}

**a.** Where is the maximum value $u(x,t)=M(T,L)$ achieved?

**b.** Where is the minimum value $u(x,t)=m(T,L)$ achieved?

**c.** Verify the maximum and minimum principle.