Find the maxima, minima, and saddle points of the function f(x,y) = (x2 - 1)(x+2).

To find the maxima, minima, and saddle points of the function f(x,y) = (x^2 - 1)(x+2), we first observe that the function only depends on x. We can write it as f(x) = x^3 + 2x^2 - x - 2. Step 1: Find the first partial derivatives and set them to zero to find critical points. The given function is f(x,y) = x^3 + 2x^2 - x - 2. The partial derivative with respect to x is: f_x = ()/( x)(x^3 + 2x^2 - x - 2) = 3x^2 + 4x - 1 The partial derivative with respect to y is: f_y = ()/( y)(x^3 + 2x^2 - x - 2) = 0 Set f_x = 0 and f_y = 0: 1) 3x^2 + 4x - 1 = 0 2) 0 = 0 From equation (1), we use the quadratic formula x = -b ± sqrt(b^2 - 4ac)2a to find the values of x: x = -4 ± sqrt(4^2 - 4(3)(-1))2(3) x = -4 ± sqrt(16 + 12)6 x = -4 ± sqrt(28)6 x = -4 ± 2sqrt(7)6 x = -2 ± sqrt(7)3 The critical points are lines where x = -2 + sqrt(7)3 and x = -2 - sqrt(7)3. Step 2: Find the second partial derivatives. f_xx = ()/( x)(3x^2 + 4x - 1) = 6x + 4 f_yy = ()/( y)(0) = 0 f_xy = ()/( y)(3x^2 + 4x - 1) = 0 Step 3: Calculate the discriminant D(x,y) = f_xxf_yy - (f_xy)^2. D(x,y) = (6x + 4)(0) - (0)^2 = 0 Since D(x,y) = 0, the Second Derivative Test is inconclusive. However, since the function only depends on x, we can analyze the behavior of f(x) using its second derivative f''(x) = f_xx(x). Step 4: Classify the critical points using f_xx. For x_1 = -2 + sqrt(7)3: f_xx(-2 + sqrt(7)3) = 6(-2 + sqrt(7)3) + 4 f_xx(-2 + sqrt(7)3) = 2(-2 + sqrt(7)) + 4 f_xx(-2 + sqrt(7)3) = -4 + 2sqrt(7) + 4 = 2sqrt(7) Since 2sqrt(7) > 0, the critical line x = -2 + sqrt(7)3 corresponds to a local minimum. For x_2 = -2 - sqrt(7)3: f_xx(-2 - sqrt(7)3) = 6(-2 - sqrt(7)3) + 4 f_xx(-2 - sqrt(7)3) = 2(-2 - sqrt(7)) + 4 f_xx(-2 - sqrt(7)3) = -4 - 2sqrt(7) + 4 = -2sqrt(7) Since -2sqrt(7) < 0, the critical line x = -2 - sqrt(7)3 corresponds to a local maximum. Step 5: Calculate the function values at these critical lines. At the critical points, 3x^2 + 4x - 1 = 0, which means 3x^2 = 1 - 4x. We can simplify f(x) = x^3 + 2x^2 - x - 2 using this relation: f(x) = x(x^2) + 2x^2 - x - 2 f(x