(5) Let u(x, t) be a solution of the following Neumann problem for the heat equation: ut(x, t) = Uzx(x, t) for x (-77, 7) and t > 0, Uz(-7,t) = 4x(, t) = 0 for t > 0, u(x,0) = f(x) for 3 (-1,1]. Assume that fe Cl(T) and u(,t) e Cl(T) for each t > 0. Also suppose f satisfies 1 favg 27 {"s(x) dx = 0. - ["u(e,t) dx. (a) Consider the quantity A(t) Show that A'(t) = 0. Conclude that A(t) = 0 for all t > 0. HINT: Differentiate under the integral sign and use the equation for u. (b) Consider the quantity Et) 1 L" u(e,t)? dr. -- Prove, using (a) and Wirtinger's inequality, that E'(t) 0, Uz(-7,t) = 4x(, t) = 0 for t > 0, u(x,0) = f(x) for 3 (-1,1]. Assume that fe Cl(T) and u(,t) e Cl(T) for each t > 0. Also suppose f satisfies 1 favg 27 {"s(x) dx = 0. - ["u(e,t) dx. (a) Consider the quantity A(t) Show that A'(t) = 0. Conclude that A(t) = 0 for all t > 0. HINT: Differentiate under the integral sign and use the equation for u. (b) Consider the quantity Et) 1 L" u(e,t)? dr. -- Prove, using (a) and Wirtinger's inequality, that E'(t)