Consider the functionf(x, y) = xy2x2 + y2 , (x, y) = (0, 0),0, (x, y) = (0, 0).(a) Use a

Chapter 14, Problem 15

(choose chapter or problem)

Consider the functionf(x, y) = xy2x2 + y2 , (x, y) = (0, 0),0, (x, y) = (0, 0).(a) Use a computer to draw the contour diagram for f.(b) Is f differentiable for (x, y) = (0, 0)?(c) Show that fx(0, 0) and fy(0, 0) exist.(d) Is f differentiable at (0, 0)?(e) Suppose x(t) = at and y(t) = bt, where a and bare constants, not both zero. If g(t) = f(x(t), y(t)),show thatg(0) = ab2a2 + b2 .(f) Show thatfx(0, 0)x(0) + fy(0, 0)y(0) = 0.Does the chain rule hold for the composite functiong(t) at t = 0? Explain.(g) Show that the directional derivative fu (0, 0) existsfor each unit vector u . Does this imply that f is differentiableat (0, 0)?