4 Let T Ri R Be Defined By F X Y Z X2 2y 32 A Explain Why We Can Always Find Local Extrema For F A Y Z I 1 (61.07 KiB) Viewed 17 times
4. Let T:RI R be defined by f(x, y, z) = x2 + 2y? +32. (a) Explain why we can always find local extrema for f(a, y, z) in the region R:= {(x, y, z): X 3y and x? + y2 + 2? <5}. (b) Use the Kuhn-Tucker method to find the local extrema of $(x,y, 2) in R. (25 marks 5. (a) Let S be a convex subset of R" and f: SR. Define what is understood by saying that S is convex and what is understood by saying that / is concave. (b) Give an example of a function g: R+R which is both quasi-convex and quasi-concave but neither convex or concave, (c) The 3-good Stone-Geary utility function, U(x,y. is given by U (..) = by log(x-01) + by log(y - c) + bs log( - ) where by, baby, c.0.cs are positive constants, 2 >> and = e. Show that the Stone-Geary utility function is quasi-concave. (15 marks)
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is given by U (..) = by log(x-01) + by log(y - c) + bs log( - ) where by, baby, c.0.cs are positive constants, 2 >> and = e. Show that the Stone-Geary utility function is quasi-concave. (15 marks)