Question

16. xyty Let f(x, y) = x3 + xy + y}, g(x, y) = x3 a. Show that there is a unique point P= (a,b) on 9(x,y) = 1 where fp = 1V9p for some scalar 1. b. Refer to Figure 13 to determine whether $ (P) is a local minimum or a local maximum of f subject to the constraint. c. Does Figure 13 suggest that f(P) is a global extremum subject to the constraint? 2 0 -3 -2 -5 X -2 0 2 Rogawski et al., Multivariable Calculus, 4e, © 2019 W. H. Freeman and Company and graph of the FIGURE 13 Contour map of f (x, y) = 2x3 + xy + y3 g(x, y) = x3 – xy + y2 = 1. constraint

Answer 1

9/ tere f( т, ) - 3 try ty? , 11, 9)13- 4 ч.) vk = [* * 1 аяз, 12 ) од - Гра 2 - 4 - 1 be be Now I f => PAR 3°ч, +29] => [3--3, -1 + зу" 2° +4 а (2х-4) x+2y = A ( 29-) я" +9 27-9 4. +24 А2 - 5. Яг \v, эм hutzke х+28 --- дух =) (272 +9) (3-2-) - +2+)(т-) * *-1° +зу°-т 2° - унтчи? - п, 41-63 439-2 -2 - - Үм ( .

q2y? 312 21 Here (1,1)= 1 Here we get GA? 4 3y²-240²t24 220 3 and gen) xy of y we can see it hat satrs thes and y also set isofred in eanul, encept there is no other point that satisfres both equation 2 PCI; I) is uarque 6) As PCI; I) is unique pont, as the point p are on both of and g. And Referring to the graph if can be easily seen that f(1,1)=g.at • at P locally manmum with respect to gan yg = n3-a yay? = beet not absolute manmelm, / IS 1 e) As it can be noot easily seen from the graph point plus locally marmur but not globally, as. from graph It can be seen that of CP) e f (8) for 8 hp =) Prs not global entremum