How do I solve this?

1.

1) a) (Ex. 3.1.12 (b), 3pts) Show that T(x, y, t) = e-kt (cos(x) + cos(y) ) satisfies the two dimensional heat equation OT k + = ay2 at b) (Ex. 3.1.29 (a), 3pts) Let f, g be C2-functions of one variable and set o(x, t) = f(x - t) +g(x + t). Show that o satisfies the wave equation a2 p/at2 = 220/212.2) Determine the second order Taylor expansion of the following functions f about the point (0, 0). Using a calculator, compute the value f (0.1, 0.1) and compare it to the value of the second order Taylor polynomial at (0.1, 0.1). a) (Ex. 3.2.4, 4pts) f(x, y) = 1/(x2 + y2 + 1). b) (Ex. 3.2.6, 4pts) f(x, y) = e - cos(xy).3) Find the critical points of the following functions and determine if they are local max- ima, local minima or saddle points. a) (Ex. 3.3.6 4pts) f(x, y) = x2 - 3xy + 5x - 2y + 6y2 + 8. b) (Ex. 3.3.10 6pts) f(x, y) = y + xsin(y). c (Ex. 3.3.17, 5pts) f(x, y) = 8y3 + 12x2 - 24xy.(Ex. 3.3.34) Let f(x, y) = 5yet - ex - 15.a) (3pts) Show that f has a unique critical point and that this point is a local maximum for f. b) (3pts) Show that f is unbounded on the yaxis and thus has no global maximum