Question

ANSWER ALL PLEASE

9 L+ 4(x 5 - (- -- - +++ a) sketch flx,y) b) Draw the level curves f(x, y) = for k=2,100,-1,-2 c) Compute f (3,3). what point represents this part b) ? computations, what the signs (²) of fx 13,3), ty (3,3), *** (3,3), tyg (3, 3)? ® Sketch the domain of f(x,y) = 5x point d) Without and + x-g² page + lu (9-x+)

Answer 1

(1) Here is the graph
(x, y, f(x, y))


-0 10 1 1 T 2

b.

-5 4. k=2 3 27 k=1 -k=0 k=-1 k=-2 1- 2 3

c. We have which lies in a level curve

-5- -4 k=2 f(3,3) 3 -2 k=1 -k=0 k=-1 k=-2 -1- 2 3

d. Remember that the first derivate says if the function is increasing/decreasing and the second derivate talks about the concavity. Therefore,

$\begin{matrix} f_x(3,3) & -\\ f_y(3,3) & 0 \\ f_{xx}(3,3) & -\\ f_{yy}(3,3) & - \end{matrix}$.

(2) For the square root term we need $x- y^2 \geqslant 0$ or $x\geqslant y^2$ . Also, for the term $\frac2{x^2 + y^2}$ we need that are not zero at the same time. For the third term to be defined we need or . These are the corresponding regions,

--- 2- -2 2 -2 -4-

and their intersction is

2 2 -2