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 — To find the maxima, minima, and saddle points of the function f(x,y) = x^2 + xy + 3x + 2y + 5, we follow these steps: Step 1: Find the first partial derivatives and set them to zero to find critical points. The given function is f(x,y) = x^2 + xy + 3x + 2y + 5. The partial derivative with respect to x is: f_x = ()/( x)(x^2 + xy + 3x + 2y + 5) = 2x + y + 3 The partial derivative with respect to y is: f_y = ()/( y)(x^2 + xy + 3x + 2y + 5) = x + 2 Set f_x = 0 and f_y = 0: 1) 2x + y + 3 = 0 2) x + 2 = 0 From equation (2): x = -2 Substitute x = -2 into equation (1): 2(-2) + y + 3 = 0 -4 + y + 3 = 0 y - 1 = 0 y = 1 The only critical point is (-2,1). Step 2: Find the second partial derivatives. f_xx = ()/( x)(2x + y + 3) = 2 f_yy = ()/( y)(x + 2) = 0 f_xy = ()/( y)(2x + y + 3) = 1 Step 3: Calculate the discriminant D(x,y) = f_xxf_yy - (f_xy)^2. D(x,y) = (2)(0) - (1)^2 = 0 - 1 = -1 Step 4: Apply the Second Derivative Test to the critical point. At (-2,1): D(-2,1) = -1 Since D(-2,1) < 0, the critical point (-2,1) is a saddle point. The final answer is Saddle point at (-2,1). Got more? Send 'em