Question 15: The unit vector along which the rate of change of the fu…

• Consider the function $f : \mathbb{R}^2 \to \mathbb{R}$ defined as

  $f(x, y) = x^2y + xy^2$

• The tangent plane to the function $f$ at the point (1,1) is given by $T_1$. Let $A_1$ denote the affine subspace of $\mathbb{R}^3$ formed by the set of points on $T_1$. Let $V_1$ denote the corresponding vector subspace associated with $A_1$.

• Let $A$ denote the Hessian matrix of the function $f$ at the point $(1, 1)$.

• Consider the affine subspace of $\mathbb{R}^3$ given by

  $A_2 = \{(x, y, z) \mid x - y + z = 1, \text{ where } x, y, z \in \mathbb{R}\}$

  Let $V_2$ denote the corresponding vector subspace associated with $A_2$.

Using the above information answer the given subquestions.

The unit vector along which the rate of change of the function ƒ at (1, 1) is the maximum is