1. (10 marks) Given that a, b, c > 0 find the maximum of f(...

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1. (10 marks) Given that a, b, c > 0 find the maximum of f(x,y, = raybe subject to the condition that x + y + 2 = 1 where x, y, 2 > 0. 2. (10 marks) Determine whether the vector field F = - - is conservative or not. If so or otherwise evaluate l' where r is the curve of intersection of the surfaces z = In(1 + x) and y = x, directed from (0,0,0) to (1, 1, ln 2). 3. (10 marks) The composite function 2 = f(x, y) has the following structure: f(x,y =F(hi(x,y))+G(h2(x,y)), = where F and G are arbitrary, single variable functions. (a) (7 marks) Ignoring any arbitrary constants, determine the functions h1 (x, y) and h2(x,y) if f satisfies the equations of = 2F' (hr(x,y)), Ox of Ox y of by = 2G'(h2(x,y)), where the primes denote differentiation with respect to the respective single variable arguments of F and G. (b) (3 marks) Show that 2 = f(x, y) satisfies the differential equation 4. (10 marks) Show that, for a > 0 and using a suitable change of variables, /o / /o FOC (x² + 2 + + 2 2 da dy dz = 2(m + TT 3)a² 5. (10 marks) The C² function = + w = T defines w as the implicit function, w = f(x,y,z). Determine values 20 and wo that satisfy the level set equation F (0, 1, 20, wo) = T. What is the maximum slope of the implicit function f at the point (0,1,20) and in what direction, u = (a,B,), is that slope found?

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