Use Lagrange multipliers to find the maximum and minimum values of the function

Question

Use Lagrange multipliers to find the maximum and minimum values of the function f subject to the given constraints g and h. f(x, y,z) yz +4zy, subject to g: ry+1 0 and h:y +422-8 0 Vf = ??g + ??h. Write out the three Lagrange conditions, i. e. Type 1 for A and j for y and do not rearrange any of the equations a) (i) Lagrange condition along x-direction: Lagrange condition along y-direction: Lagrange condition along z-direction: Find the exact value of l and use this to rewrite the Lagrange condition in the y-direction. (ii) Lagrange condition along y-direction: (iii) Write out the condition along the z-direction after eliminating J. (Rearrange to avoid any divisions in your equation.) Lagrange condition along z-direction: (iv) Use the relation found in) and one of the constraints to find Find all the points at which f has extreme values subject to the constraints g and h as above b) (i) Which combination of the following sets contains these critical points? (1/2, 2,1), (-1/2,-2,-1)) i How many points are there in total? 02 04 0 16 c) Find maximum and minimum values of f fmax = min

Explanation / Answer

a) i) 4y=ly partially differentiate all the equations wrt x and substitute it in the Lagrange conditions

z+4x=lx+j(2y)

y=j8z

ii) l =4

z=j*2y

iii)4z^2=y^2

iv)4z^2

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