Hi, I have the following question:

Consider the function f(x,y,z)=x^3+y^4-z^3

1) Find the points on the surface f(x,y,z)=1 that have horizontal tangent plane (tangent plane parallel to the xy plane).

This is the question I am unsure of.

However I have:

grad f=(3x^2)i+(4y^3)j-(3z^2)k

2) Compute the directional rerivative at point (0,1,1) in the direction of u=i+j+k

I found this answer to be 1/sqrt(3)

3) Suppose G(x,y)=f(x,y,0)=x^3+y^4 describes the profit of a company as a measure of two variables. What should be your best strategy to modify y as a function of x in order to maximise the profit if you know that initially x=1, y=1

I got the path of maximal increase to be y=(y^4)/(3x^2) + (2/3)

However I am unsure what to say about modifying y.

Cheers.