Consider the problem of finding extrema of f (x, y,z) = x 2 + y2 subject to the

Chapter 4, Problem 42

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Consider the problem of finding extrema of f (x, y,z) = x 2 + y2 subject to the constraint z = c, where c is any constant. (a) Use the method of Lagrange multipliers to identify the critical points of f subject to the constraint given above. (b) Using the usual alphabetical ordering of variables (i.e., x1 = x, x2 = y, x3 = z), construct the Hessian matrix H L(; a1, a2, a3) (where L(l; x, y,z) = f (x, y,z) l(z c)) for each critical point you found in part (a). Try to use the second derivative test for constrained extrema to determine the nature of the critical points you found in part (a). What happens? (c) Repeat part (b), this time using the variable ordering x1 = z, x2 = y, x3 = x. What does the second derivative test tell you now? (d) Without making any detailed calculations, discuss why f must attain its minimum value at the point (0, 0, c). Then try to reconcile your results in parts (b) and (c). This exercise demonstrates that the assumption that det g1 x1 (a) g1 xk (a) . . . ... . . . gk x1 (a) gk xk (a) = 0 is important.