Exam practice question to which I don't have an answer. Would be greatful for any help.

Question.

Consider the function

defined by

,

and let P=(2,1,3).

(i) Find the rate of change of f(x,y,z) at the point P in the direction

.

(ii) Write down the direction of maximum increase of f at P, and the rate of maximum increase. Show that P is on the surface

and write down the equation of the tangent plane to the surface at P.

(iii) There is another point Q on the surface

where the tangent plane is parallel to the tangent plane at P. Find Q.

Answer.

(i) first, I find partial derivatives of f at P

Let

be the angle between the above vector (6,-10,2)^T and the unit vector v=(1,-1,1)^T. Then I can find alpha:

Then the rate of change of

in the direction of v=(1,-1,0)^T is just the length of the projection of this vector on the direction of v, which can be calculated as