2. Multivariable Calculus
(a) Given $f(x, y) = \cos xy + 3x^2e^{-y}$,
Show the second partial derivatives $f_{xy} = f_{yx}$.
(b) The linear approximation to a function at point (a,b) is given by
$f(a,b) \approx L(x, y) = f(a,b) + f_x(a, b) (x - a) + f_y(a, b) (y - b)$.
(i) Find the linear approximation of the function $f(x, y) = e^{2x^2+3y}$ at (0,0) and
(ii) use it to approximate $f (0.01, -0.02)$.
(c) Find and classify the critical points of:
$f(x,y) = 2x^4 - 8xy + y^2 - 3$.