The following scenario describes the temperature u of a rod at position x and time t.
Consider the equation u_t = u_{xx}, 0 < x < 1, t > 0, with boundary conditions u(0,t) = 0 , u(1,t) = 0. Suppose

u(x,0) =
6 sin(4 pi x)

What is the maximum temperature in the rod at any particular time. That is,