A cold steel bar of length

L = 1 is held in mid-air by two posts which can be considered as a heat leak to the lab with a reference temperature Tref = 6. Initially (time t = 0), the temperature distribution across the steel bar is given by

T(t = 0; x) = -3x + 17x^2- 10x^3 0<= x <= 1

The question which i am struggling on is:

Q) Show that

T(x; t) = Tref is a solution of the heat equation including the boundary conditions. This is the state the steel bar will approach for large times. How much deviates the average temperature from Tref at t = 0:5?

Any help will be great.

Thanks