Question

This extreme value problem has a solution with both a maximum value and a minimum value. Use Lagrange multipliers to find the extreme values of the function subject to the given constraint.

f(x, y, z) = x² + y² + z²;    x⁴ + y⁴ + z⁴ = 7

Maximum Value:

Minimum Value:

This extreme value problem has a solution with both a maximum value and a minimum value. Use Lagrange multipliers to find the extreme values of the function subject to the given constraint. f(x, y, z) = x2 + y2 + z2; x4 + 4 + 24 = 7 maximum value minimum value X

Answer 1

Solution: Let (x, y, z) = x+y+z2 g[x,y,z)= x' + y +z=7 The Lagrange equation Vg 2Vf gives f = {2x, 29, 22} and Vg = 4x 4p; 4*} Now Yg = IV = 4x = a[2x)= 2^(2x - aj=0=r=0 or x=+ 4y= = a[2y) - 2y/2y - aj=0=y=0 or y =+ 只 42= = (22) = 22[22 - aj=0=z=0 orz =+ 22 la x = y = 2 = +, 2 Now plugging into constraint equation: x+y+' =7 a as a + + 7号 7= =, since I cannot take negative values 4 4 3 128 114 * = Y = = 4, 3 = self , - 停一停, 停停,停停 [14 3 14 14 114 F14 114 14 114 14 f3f3 3 13 14 14 14 At all above points, f(x, y, z)==+ += = 14 3 3 3 Wies = y= 02 = +7 When x=2= = 0=r = +7 When z = y=0)=x=377 (x,y,z)=(0.0.47) (0.0-37) (0.37.0(0-37.0)(47,0,0),(-47.00) : At all above points, f (x,y,z)= 77 「14 114 14 (x,y,z)=1 (停停,停停停, 停停停 3 3 Hence Maxima = 14 at 114 114 14 14 14 114 14 14 一号) 3 3 11/3 3 and inpute= lock Lay.z)=(0.0.47) (0.0-575) (0.37.0) (0.47.0) {7,0.0),(-57.0.01] Tha Maxima mentioned above is a Local Maxima and no global Maxima occurs as f(x,y,z) is unbounded from above 114 3 114 . . 114 3 14 3 3