49 please 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) If 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 = bL K¹-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. Showad that the maximum production occurs whenilgillum ogn (1-9)and for K =287L=арm-PSulev moninim sai bart of28. Referring to Exercise 27, we now suppose that the pro-duction is fixed at bLªK¹-a Q, where Q is a constant.What values of L and K minimize the cost functionC(L, K) = mL + nK?SIROTIXS 50 bar17 US-FE157329. Use Lagrange multipliers to prove that the rectangle withmaximum area that has a given perimeter p is a square.t=n30. Use Lagrange multipliers to prove that the triangle withmaximum area that has a given perimeter p is equilateral.Hint: Use Heron's formula for the area:- y)(s — z)x)(sy)(sz) burt 2E-TSMilupani adiwhere s = p/2 and x, y, z are the lengths of the sides.A = √√√s(sx) (s31-43 Use Lagrange multipliers to give an alternate solution tothe indicated exercise in Section 14.7.31. Exercise 4133. Exercise 4335. Exercise 4537. Exercise 4739. Exercise 4932. Exercise 4234. Exercise 4436. Exercise 4638. Exercise 4840. Exercise 5041. Exercise 5143. Exercise 5544. Find the maximum and minimum volumes of a rectangularbox whose surface area is 1500 cm² and whose total edgelength is 200 cm.45. The plane x + y + 2z = 2 intersects the paraboloidz = x² + y² in an ellipse. Find the points on this ellipsethat are nearest to and farthest from the origin.46. The plane 4x-z² = x² + y² in an ellipse.(a) Graph the cone and the plane, and observe the ellipticalnolaupe intersection.3y + 8z = 5 intersects the cone(b) Use Lagrange multipliers to find the highest and lowestnisl points on the ellipse.42. Exercise 52CAS 47-48 Find the maximum and minimum values of f subject tothe given constraints. Use a computer algebra system to solvethe system of equations that arises in using Lagrange multipliers.(If your CAS finds only one solution, you may need to use addi-tional commands.)47. f(x, y, z) = ye¹-²; 9x² + 4y² + 36z² = 36, xy + yz = 148. f(x, y, z) = x+y+z; x² - y² = z, x² + z² = 449. (a) Find the maximum value off(x1, x2,...,xn)= X1X2. Xngiven that x₁, x2, ..., x, are positive numbers andhtlgX1 + x₂ + ... + x = c, where c is a constant.love (b) Deduce from part (a) that if x₁, x2, ..., xn are positive$ 60numbers, thenX1 X2 Xn <X₁ =451144pThis inequality says that the geometric mean of n num-bers is no larger than the arithmetic mean of thenumbers. Under what circumstances are these twoai√Σ a²Σαto show thatx₁ + x₂ + ... + xnw nottul means equal?dong sulav omonis dead to dos #1-2101 igaldu Σ7-1x²50. (a) Maximize Σ₁ x/y, subject to the constraints diodΣ=1x? = 1 and Σ-₁ y² = 1. sitt bad of oilghtom(b) Putnand yi=bi√Σ b²Σ ab; < Σ a {Σ bfor any numbers a₁,..., an, bi, ..., bn. This inequalityis known as the Cauchy-Schwarz Inequality.

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- (a) Find the maximum value of f(x₁, x2,...,xn) = √√√x₁x₂x₂ given that x₁, x2,...,x, are positive numbers and x₁ + x₂ + + x = c, where c is a constant. (b) Deduce from part (a) that if x₁, x2,...,xn are positive numbers, then x₁ + x₂ + ... + xn n DIRVO This inequality says that the geometric mean of n num- bers is no larger than the arithmetic mean of the numbers. Under what circumstances are these two ou means equal?on sulavamentes asi to dos AT-E (a) Man √√X1 X2 X =