Finding critical points of the function

Question

Assignment:

Question 1. Find the critical points of the function

f(x, y) = 4xy² -  4x² -  2y²

and classify them as local maxima, minima or saddles or none of these.

Question 2. The surface is defined by

z = 3x² + 2y² - 3

Find the equation of the tangent plane to the surface at the point (1, 1, 2).

Question 3.

Find the directional derivative of the function f(x, y) = x²e^y  at the point (1,0) in the direction i + 2j.
Find the gradient vector of f (x, y).  In which direction, (i + 2j) or (2i + j), does f(x, y) change more rapidly? Explain.

Provide complete and step by step solution for the question and show calculations and use formulas.