3 inches Matched structed from 288 square box with the largest possible volume 6 inches 9 inches CONCEPTUAL INSIGHT Figure 8 An alternative to the method of Lagrange multipliers would be to solve Example by means of Theorem 2 (the second-derivative test for local extrema) in Section That approach involves solving the "(62-xy Variables, 2 = 2x + 3y Then we would eliminate z in the volume function to obtain a function of two 162 - xy variables: V ( x, y ) = x 2x + 3y The method of Lagrange multipliers allows us to avoid the formidable tasks calculating the partial derivatives of V and finding the critical points of V in or to apply Theorem 2. Exercises 7.4 10. Maximize f(x, y) = 25 -x2-2 Skills Warm-up Exercises subject to 2x + y = 10 W In Problems 1-6, maximize or minimize subject to the constraint without using the method of Lagrange multipliers; instead, solve B In Problems 11 and 12, use Theorem I to explain why n the constraint for x or y and substitute into f(x, y). (If necessary, or minima exist. review Section 1.3). 11. Minimize f(x, y) = 4y - 3x aint. 1. Minimize f(x, y ) = >+ xy+ 2 subject to 2x + 5y = 3 subject to y = 4 12. Maximize f(x, y) = 6x + 5y + 24 2. Maximize f(x, y) = 64 + x2 + 3 xy -yz subject to 3x + 2y = 4 subject to x = 6 3. Minimize f(x, y) = 4xy Use the method of Lagrange multipliers in Problems subject to x - y = 2 13. Find the maximum and minimum of f(x, y) = 4. Maximize f(x, y) = 3xy x2 + 12 = 18. subject to x + y = 1 14. Find the maximum and minimum of f(x, y) 5. Maximize f(x, y) = 2x + y subject to x2 + y2 = 25. subject to x2 + y = 1 sol 6. Minimize f(x, y) = 10x - yz 15. Maximize the product of two numbers if their subject to *2+ y2 = 25 16. Minimize the product of two numbers if their be 10. A Use the method of Lagrange multipliers in Problems 7-10. 7. Maximize f(x, y) = 2xy c 17. Minimize f(x, y, z) = x2 + 2+ z subject to x + y = 6 subject to x + y + 3z = 55 8. Minimize f(x, y) = 6xy 18. Maximize f(x, y, z) = 300 - x2 - 2yz subject to y - x = 6 subject to 4x + y + z = 70 9. Minimize f(x, y) = x2 + yz 19. Maximize f(x, y, z) = 900 - 5x2 - yz subject to 3x + 4y - 25vogeleditors subject to x + y + z = 34 20. Minimize f(x, y, z) =x2+ 4y2+ 22 subject to x + 2y + z = 10