The figure below shows the level curves of a function $f(x,y)$ around a maximum or minimum, $M$. The $x$- and $y$-axes are along the bottom and left edge of the figure, respectively.

One of the points $P$ and $Q$ has coordinates $(x_1,y_1)$ and the other has coordinates $(x_2, y_2)$. Suppose $b > 0$ and $c > 0$. Consider the two linear approximations to $f$ given by $$f(x,y) \approx a+b(x-x_1)+c(y-y_1),$$ and $$f(x,y) \approx k+m(x-x_2)+n(y-y_2).$$

(a) What is the relationship between the values of $a$ and $k$?

(Give your answer as an equation.)

(b) What are the coordinates of $P$?
? (x1,y1) (x2,y2)

(c) Is $M$ a maximum or a minimum? ? maximum minimum

(d) What can you say about the sign of the constants $m$ and $n$?
$m$ ? > >= = < <= 0, and
$n$ ? > >= = < <= 0.