Question

Consider the surface defined by 2 = f(x,y), where f(x, y) = (x + y2 - 1)(x + y - 4). (a) In three separate diagrams draw the level sets of the function at C=2, C = 4, and C= 6. State the coordinates of any isolated points and the radii of any circles that make up these level sets. (Hint: To get an idea of what the surface looks like it might help to look at the curves f(0,y) and f(t,0)). (b) Write down the domain and range of f(x,y). (c) For what values of (1,y) does the function f(x,y) obtain: (i) Its minimum value? (ii) A local maximum?

Answer 1

The given problem is to find the range and domain,,minimum value and local maximum...all the steps are clearly written in the pic....if you havea ny doubt ask in the comment section...THANK YOU :-)
z=f(a,h), where fcn,y) = (x² + y²-1) (x² + y24) a) Graph: be,y) at (1,1) 6) Domain & range of eig. frey) - (x² + y²-1) (x+y²-4) here fin,y) es de frined, so, their is no point (x,y) for which f(x,y) is undefined No daman Range: domain is undefried, so fluy) satisfies everywhere, so Range of fracy i R. = f.Ge, y) = (x+y2-1) (x² + y²-4) in Critical pornts: Ex(x, y) = 0; ty (ny)=0 fx (n,y) = (2x) (2x) = 0 x=0 y (r, y) = (24) (24) o yao.

for Cany) & (482) = 8x = 0 (0,0) fxx (2,4)= 0 at (0,0) So, it is inconclusive minmum value Not defined y los al momenbucom o Dahere critical points en hath cord & the tocat nominen in local maximum is at where critical points exhts. That is (0,0) is the local maximur