u'; 2. (4 pts) Find the global maximum and global minimum values . . b (if they exist) of JJ?' + yz in the region as + y < 1. If there is no global max. Justify why. If there - is no global minimum, justify why. 3. (4 pts) F md the inaxunum and minimum value of f(J?, y. 2, f) :: .r+y+z+t under the constraint ;z:'2+y2+z2+t2 = 1 4. (4 pts) Find the. maxinnun of f('r. = 243: -- it? + 16y -- 23/2 when x2 + 2y2 : 44. What is the approximate change in the maximum value of f if the constraint is changed to 172 + 23/2 = 45? 5. (4 pts) An individual purchases quantities a, b. and, c of three different commodities whose prices are p, 7', respectively. The consumer spends 'm dollars, where m > 2p, and the utility function is given by U(a, b, c) = a + ln(bc) Show that when utility is maximized the expenditure on each of the second and the third good is equal to p. Find it, bi and c When utility is maximized. (assume Lagrange multipliers gives the max solution) q, and 6. (4 pts) The function f(x, y, z) = (.v-4)2+(y--4)2+(z- %)2 has a minimum subject to the constraint sc2+y2-z : 0. Find the point (a, b, c) where f attains the minimum value subject to the constraint. (you may assume the method of Lagrange multipliers gives the minimum) 3 -1 4 1 1 1 7. (4pts)LetA= 5 10 -12 ,3= 1 3 9 12 0 7 1 4 16 (a) (2 pts) Find A --- 3B. 3; 3 (b) (2 PtS) If v = [ y and b = 2 , then setting Av = b gives us 'a system of 3 equations in the 3 z -1 unknowns x, y, and 2. Write the system of equations (you don't need to solve it).2. (4 pts) Find the global maximum and global minimum values (if they exist) of 2132 + If there is no global max, justify Why. If there is no global minimum, justify why. 3. (4 pts) Find the maximum and minimum value of f (;17, y, 2 ya in the region it + y g l. ,t) = x+y+z+t under the constraint .'r,2+y2+t2 +t2 : 1. flat, y) 2 24m - :B2 + ltiy - 2y2 when 332 + 2y2 2 44. What is the approximate of f if the constraint is changed to 1122 + 2y2 2 45? 4. (4 pts) Find the maximum of Change in the maximum value 5. (4 pts) An individual purchases quantities a, b, and, C of three different commodities whose prices are p. q. and 7", respectively. The consumer spends m dollars, where m > 2p, and the utility function is given by U(a, b, c) = a + ln(bc) ' Show that when utility is maximized, the expenditure on each of the second and the third good is equal to 1). Find a, b, and c When utility is maximized. (assume Lagrange multipliers gives the max solution) .2 2--,.. 6. (4 pts) The function f(:L', y, z) = (z1:--4)2+(y-4)2+(Z-)2 has ammimum subject to the consti mm in +y c .. 0. Find the point (a, b, c) Where f attains the minimum value subject to the constraint. (you may assume the method of Lagrange multipliers gives the minimum) 3 -1 4 1 1 1 7. (4pts)LetA= 5 10 -12 ,B= 1 3 9 12 0 7 1 4 16 (a) (2 pts) Find A -- 3B. 3 . l, and b : 2 , then setting AV 2 b gives us a system of 3 equations in the 3 -1 . unknowns at y and 2. Write the system of equations (you don't need to solve it). 7 ' (b) (2 pts) If v = NQQE-g\f2. (4 pts) Find the global maximum and glob al minimum values (if they exist) of zc2 + y2 in the re If there is no global in gion x + y g 1, ax, Justify Why. If there is no global minimum, justify why. 3. (4 pts) Find the maximum and minimum value of f(:1:, y, z, t) = 33+y+z+t under the constraint x2+yz+z2+t2 :: 1. 4. (4 pts) Find the maximum of f(:u, y) = 2437 - 552 + 16y -- 2y2 when {1:2 + 2y2 : 44. What is the approximate Change in the maximum value of f if the constraint is changed to 5132 + 2y2 : 45? 5. (4 pts) An individual purchases quantities a, b , and, c of three different commodities whose prices are p, q, and 7', respectively. The consumer spends m doll ars, where m > 2p, and the utility function is given by U(a, b, c) = a + ln(bc) Show that when utility is maximized, the expenditure on each of the second and the third good is equal to p. Find a, b, and 0 when utility is maximized. (assume Lagrange multipliers gives the max solution) 6. (4 pts) The function f(x, y, z) = (at-4)2+(y-4)2+(z-%)2 has a minimum subject to the constraint x2+y2-z = O. Find the point (a, b, c) Where f attains the minimum value subject to the constraint. (you may assume the method of Lagrange multipliers gives the minimum) 3 -1 4 1 1 1 7. (4pts)LetA= 5 10 ---12 ,B=: 1 3 9 12 0 7 1 4 16 (a) (2 pts) Find A --- 3B. 3; 3 (b) (2 pts) If v : y and b : 21 , then setting AV = b gives us 'a system of 3 equations in the 3 unknowns sc, y, and 3. Write the system of equations (you don't need to solve it)