1) Find the gradient vector f for f(x, y, z) = x2 - 2xy + 3z² at the point P(1, 3, 2). 1)

2) Given f(x, y, z) = x3 - 3xyz + z4, (a) in what direction is f increasing the most rapidly at 2) the point (1, -1, 1)?
(b) What is the rate of increase of f in that direction at that point?

3) Use the method of Lagrange multipliers to find the extreme values of 3x - 4y + 12z on 3) the spherical surface with equation x² + y² + z² = 1.

4) Use the method of Lagrange multipliers to find the extreme values of the expression

x^2y^2z^2 on the spherical surface with equation x2 + y2 + z2 = 3 and the points on the surface at which these extrema occur.

5) Given f(x, y, z) = x^2y^3z^6, in what direction is f(x, y, z) increasing the most rapidly at 5) the point P(1, -1, 1)? What is its rate of increase in that direction?