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
Step 1: Find the first-order partial derivatives f_x and f_y. The given function is: f(x,y) = 2x^2 + 3xy + 4y^2 - 5x + 2y Differentiate f(x,y) with respect to x (treating y as a constant): f_x = ()/( x)(2x^2 + 3xy + 4y^2 - 5x + 2y) f_x = 4x + 3y - 5 Differentiate f(x,y) with respect to y (treating x as a constant): f_y = ()/( y)(2x^2 + 3xy + 4y^2 - 5x + 2y) f_y = 3x + 8y + 2 Step 2: Find the critical points by setting f_x = 0 and f_y = 0. Set f_x = 0: 4x + 3y - 5 = 0 (1) Set f_y = 0: 3x + 8y + 2 = 0 (2) From equation (1), we have 4x + 3y = 5. From equation (2), we have 3x + 8y = -2. To solve this system of linear equations, multiply equation (1) by 3 and equation (2) by 4: 3(4x + 3y) = 3(5) 12x + 9y = 15 (3) 4(3x + 8y) = 4(-2) 12x + 32y = -8 (4) Subtract equation (3) from equation (4): (12x + 32y) - (12x + 9y) = -8 - 15 23y = -23 y = -1 Substitute y = -1 into equation (1): 4x + 3(-1) - 5 = 0 4x - 3 - 5 = 0 4x - 8 = 0 4x = 8 x = 2 The only critical point is (2, -1). Step 3: Calculate the second-order partial derivatives. f_xx = ()/( x)(4x + 3y - 5) = 4 f_yy = ()/( y)(3x + 8y + 2) = 8 f_xy = ()/( y)(4x + 3y - 5) = 3 Step 4: Calculate the discriminant D(x,y). The discriminant is given by the formula: D(x,y) = f_xxf_yy - (f_xy)^2 Substitute the partial derivatives: D(x,y) = (4)(8) - (3)^2 D(x,y) = 32 - 9 D(x,y) = 23 Step 5: Evaluate D(x,y) and f_xx(x,y) at the critical point. For the critical point (2, -1): D(2, -1) = 23 Since D(2, -1) = 23 > 0, we need to check f_xx(2, -1). f_xx(2, -1) = 4 Since f_xx(2, -1) = 4 > 0 and D(2, -1) > 0, the point (2, -1) is a local minimum. Calculate the function value at (2, -1): f(2, -1) = 2(2)^2 + 3(2)(-1) + 4(-1)^2 - 5(2) + 2(-1) f(2, -1) = 2(4) - 6 + 4(1) - 10 - 2 f(2, -1) = 8 - 6 + 4 - 10 - 2 f(2, -1) = 2 + 4 - 10 - 2 f(2, -1) = 6 - 10 - 2 f(2, -1) = -4 - 2 f(2, -1) = -6 Summary of results: • Local minima: There is a local minimum at (2,-1) with a value of f(2,-1) = -6. • Local maxima: There are no local maxima. • Saddle points: There are no saddle points. Got more? Send 'em!