Student is asked to investigate how the function $f$, given by
$f(x, y, z) = z - (x^4 + y^2 - xy)$
behaves at the point P(1, 2, 3).
3.1 Obtain $\nabla f$, and hence obtain the value of this vector at the point P.
Round components to four decimal places, where necessary.
3.2 Verify whether or not $\nabla f$ in 3.1 is solenoidal.
3.3 Obtain the $(x, y, z)$ coordinates of all critical points of $z = g(x, y) = x^4 + y^2 - xy$.
3.4 Determine the type of critical points of $g(x, y) = x^4 + y^2 - xy$ using second derivative test.
