Exercises 7.3 Skills Warm-up Exercises 18. f(x, y) = 15x2 - 12 - 10y In Problems 1-8, find f'(0), f"(0), and determine whether f has a local minimum, local maximum, or neither at x = 0. (If neces 19. f ( x, y ) = 8 - x2 + 12x - 12 - 2y sary, review the second derivative test for local extrema in Section 20. f( x, y ) = x2 + y2 + 6x - 8y+10 4.5 ). 21. f( x, y) = x2+ 3xy + 2y2 + 5 1 . f(x ) = 28 - 9x2+ 4 2. f(x) = 4x3 + 6x2 + 100 22. f(x, y ) = 4x2 - xy + 2+ 12 3. f(x) = 1 - X2 4. f ( x ) = - 23. f(x, y) = 100 + 6xy - 4x2 - 3y2 1 + * 2 24. f(x, y ) = 5x2 - y2 + 2y + 6 5 . f(x ) = ex 6. f(x) = et 25. f ( x, y ) = x2 + xy + y2 - 7x + 4y +9 7. f(x ) = x- x + x - 1 8. f(x) = (3x + 1)2 26. f(x, y) = -x2 + 2xy - 2y2 - 20x + 34y + 40 In Problems 9-16, find fx(x, y) and fy (x, y), and explain, using C 27. f(x, y) = ey Theorem 1, why f(x, y ) has no local extrema. 28. f (x, y ) = xy - xy2 9. f( x, y) = 4x + 5y - 6 29. f( x, y) = x3 + y - 3xy 10. f( x, y ) = 10 - 2x - 3y + x2 30. f( x, y) = 2y' - 6xy - x2 11. f(x, y ) = 3.7 - 1.2x + 6.8y + 0.2y3 + xtor 31. f(x, y) = 2x* + y2 - 12ry 12. f( x, y ) = x3 - 12 + 7x + 3y + 1 32. f(x, y) = 16xy - x4 - 2y? 13. f( x, y ) = -x2 + 2xy - y2 - 4x + 5y 33. f( x, y) = x - 3 xy2 + 62 14. f( x, y ) = 3x2 - 12xy + 12y2 + &x + 9y - 15 34. f (x, y) = 212 - 272y + 6y3 15 . f( x, y ) = yet - 3x + 4y 35. f(x, y ) = xe' + xy + 1 16 . f( x, y ) = y + y? Inx 36. f( x, y ) = ylnx + 3xy In Problems 17-36. use Theorem 2 to find the local extrema. 37. Explain why f(x, y) = x has a local extremum at infinitel many points. 17 . f(x, y ) = 12 + 8x + 12 + 25Step 2: Find fxx, f\". and f\". Step 3: Use the Second Derivative test for each critical point. Use the Second Derivative test for each critical point. Critical points Calculate d using :1 = fxx(a.b)f (1.11) Classify using 2Ml Derv Test