25 978CHAPTER 14 Partial Derivatives25. Consider the problem of minimizing the functionf(x, y) = x on the curve y² + x4 - x³ = 0 (a piriform).(a) Try using Lagrange multipliers to solve the problem.(b) Show that the minimum value is f(0, 0) = 0 but theLagrange condition Vf(0, 0) = AVg(0, 0) is not satis-fied for any value of A.(c) Explain why Lagrange multipliers fail to find the mini-mum value in this case.CAS 26. (a)33If your computer algebra system plots implicitly definedcurves, use it to estimate the minimum and maximumvalues of f(x, y) = x³ + y³ + 3xy subject to the con-straint (x - 3)² + (y - 3)2 = 9 by graphical methods.(b) Solve the problem in part (a) with the aid of Lagrangemultipliers. Use your CAS to solve the equations numer-ically. Compare your answers with those in part (a).27. The total production P of a certain product depends onthe amount L of labor used and the amount K of capitalinvestment. In Sections 14.1 and 14.3 we discussed how theCobb-Douglas model P = bLºK¹-a follows from certaineconomic assumptions, where b and a are positive constantsand a < 1. If the cost of a unit of labor is m and the costof a unit of capital is n, and the company can spend onlyp dollars as its total budget, then maximizing the produc-tion P is subject to the constraint mL + nK = p. Showthat the maximum production occurs whenvonL =apmand K =(1 - a)pn28. Referring to Exercise 27, we now suppose that the pro-duction is fixed at bL"K¹-a "=Q, where Q is a constant.

So, the solution is1,0, with the valuef1,0=1, but f1,0 cannot be the minimize f as f0,0=0, which is…

Consider the problem of minimizing the function f(x, y) = x on the curve y² + x² - x³ = 0 (a piriform). (a) Try using Lagrange multipliers to solve the problem. (b) Show that the minimum value is f(0, 0) = 0 but the Lagrange condition Vf(0, 0) = AVg(0, 0) is not satis- fied for any value of A. (c) Explain why Lagrange multipliers fail to find the mini- mum value in this case. implicitly defined