For a" : 0 the problem _ 2 saw T) has two solutions, a: = :Izl. For T 7 0, there is a unique solution. Show that the value function V0") is not differentiable at r = 0. You can assume that |:L'| is not differentiable at 0. [15 Marks] Are the following statements true or false? Justify your answers (a) If f (3:,y) has a local maximum at ((1,6) subject to the constraint g(a:,y) = c, then 9(a, b) : c. [3 Marks] (b) If f (say) has a local maximum at ((1,5) subject to the constraint g(m,y) = c, then Vf(a,, b) = 0 . [3 Marks] (c) If V f (a, b) and Vg(a, b) point in opposite directions, then (a, b) is a local minimum of f (2:, y) constrained by g(a:, y) : c. [3 Marks] Consider the function f (1:, y) : sin(:1:) sin(y). Compute the second order Taylor polynomial of f around the points (0, 0) and (arr/2, 77/2). [13 Marks] If MCI, t) = 6'\" sin(ba:) satises the heat equation at = um, nd the relationship between a and b. [18 Marks]