Exercise 14

Let $f(x,y) = 8 - x^2 - y^2$ and $g(x,y) = 8 - x^2 + 4xy - y^2$.

1. Determine $f_x$, $f_y$, $f_{xx}$, $f_{yy}$, $f_{xy}$, and $f_{yx}$.

2. Evaluate each of the partial derivatives in (a) at the point $(0,0)$.

3. What do the values in (b) suggest about the behavior of $f$ near $(0,0)$? Plot a graph of $f$ and compare what you see visually to what the values suggest.

4. Determine $g_x$, $g_y$, $g_{xx}$, $g_{yy}$, $g_{xy}$, and $g_{yx}$.

5. Evaluate each of the partial derivatives in (d) at the point $(0,0)$.

6. What do the values in (e) suggest about the behavior of $g$ near $(0,0)$? Plot a graph of $g$ and compare what you see visually to what the values suggest.

7. What do the functions $f$ and $g$ have in common at $(0,0)$? What is different? What do your observations tell you regarding the importance of a certain second-order partial derivative?