I am stuck on the last two questions could someone help me out? Consider the fun

Question

I am stuck on the last two questions could someone help me out?

Consider the function f ( x , y ) = 1/ x2 + y2 + 1. Find equations for the following level surfaces for f and sketch them, (i) f(x,y) = 1/5: (ii) f(x,y) = 1/10. Find all values of k. if any. such that the level surface f(x,y) = k consists of a single point. Why does the level surface corresponding to k = 1 the absolute maximum of f(x,y)? Is it possible to find a function for which it is true that, for all x > 0 and y > 0, fx > 0 and fy 0? If so. give an example. If not. why not?

Explanation / Answer

6)a)

i)x2+y2+1=5

x2+y2=4>>is a circle with radius 2 and centre at (0,0)

ii)x2+y2+1=10

x2+y2=9>> is a circle with radius 3units and centre at(0,0)

b)

k=1/(x2+y2+1)

x2+y2+1=1/k

x2+y2=(1/k)-1

for k=1 we get a single point (0,0)

c)

for all values of (x,y) f(x,y) will be less than 1 except at point(0,0) which gives the maximum value for the function

for k=1

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