This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
You're on a roll — Step 5: Evaluate D(x,y) and f_xx(x,y) at each critical point (continued). For (-2, -1): D(-2, -1) = 36((-2)^2 + (-2)(-1) - (-1)^2) D(-2, -1) = 36(4 + 2 - 1) D(-2, -1) = 36(5) = 180 Since D(-2, -1) > 0, we check f_xx(-2, -1): f_xx(-2, -1) = 6(-2) = -12 Since f_xx(-2, -1) < 0 and D(-2, -1) > 0, (-2, -1) is a local maximum. The function value is f(-2, -1) = (-2)^3 + 3(-2)(-1)^2 - 15(-2) + (-1)^3 - 15(-1) f(-2, -1) = -8 - 6 + 30 - 1 + 15 f(-2, -1) = -15 + 45 = 30 Summary of results: • Local minima: There is a local minimum at (2,1) with a value of f(2,1) = -30. • Local maxima: There is a local maximum at (-2,-1) with a value of f(-2,-1) = 30. • Saddle points: There are saddle points at (0, sqrt(5)) with a value of f(0, sqrt(5)) = -10sqrt(5) and at (0, -sqrt(5)) with a value of f(0, -sqrt(5)) = 10sqrt(5). What's next?