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
To find the maxima, minima, and saddle points of the function f(x,y) = x^2 + xy + y^2 + 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 + y^2 + 3x + 2y + 5. The partial derivative with respect to x is: f_x = ()/( x)(x^2 + xy + y^2 + 3x + 2y + 5) = 2x + y + 3 The partial derivative with respect to y is: f_y = ()/( y)(x^2 + xy + y^2 + 3x + 2y + 5) = x + 2y + 2 (Note: The f_y in your work was x+2, which is incorrect for the given function. We will use the correct f_y = x+2y+2.) Set f_x = 0 and f_y = 0: 1) 2x + y + 3 = 0 2) x + 2y + 2 = 0 From equation (1), we can express y in terms of x: y = -2x - 3 Substitute this into equation (2): x + 2(-2x - 3) + 2 = 0 x - 4x - 6 + 2 = 0 -3x - 4 = 0 -3x = 4 x = -(4)/(3) Now substitute x = -(4)/(3) back into the expression for y: y = -2(-(4)/(3)) - 3 y = (8)/(3) - (9)/(3) y = -(1)/(3) The only critical point is (-(4)/(3), -(1)/(3)). Step 2: Find the second partial derivatives. f_xx = ()/( x)(2x + y + 3) = 2 f_yy = ()/( y)(x + 2y + 2) = 2 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)(2) - (1)^2 = 4 - 1 = 3 Step 4: Apply the Second Derivative Test to the critical point. At (-(4)/(3), -(1)/(3)): D(-(4)/(3), -(1)/(3)) = 3 Since D > 0, we check f_xx: f_xx(-(4)/(3), -(1)/(3)) = 2 Since f_xx > 0, the critical point (-(4)/(3), -(1)/(3)) is a local minimum. Step 5: Calculate the function value at the local minimum. f(-(4)/(3), -(1)/(3)) = (-(4)/(3))^2 + (-(4)/(3))(-(1)/(3)) + (-(1)/(3))^2 + 3(-(4)/(3)) + 2(-(1)/(3)) + 5 f(-(4)/(3), -(1)/(3)) = (16)/(9) + (4)/(9) + (1)/(9) - (12)/(3) - (2)/(3) + 5 f(-(4)/(3), -(1)/(3)) = (21)/(9) - (14)/(3) + 5 f(-(4)/(3), -(1)/(3)) = (7)/(3) - (14)/(3) + (15)/(3) f(-(4)/(3), -(1)/(3)) = (7 - 14 + 15)/(3) = (8)/(3) The final answer is Local minimum at (-(4)/(3), -(1)/(3)) with value (8)/(3). 3 done, 2 left today. You're making progress.