Question

Consider the function f(x, y): = −2x+2x² - 2y + xy + x²y - y²+xy² (a) Factor the function f(x) into a product of three binomials. For example, if g(x, y) = x - x² + y + 2xy + x²y - y2 - xy2, you would rewrite the function into g(x,y) = (xy) (y-1)(x+1) (b) Find all the critical points and classify them as minima, maxima, or saddle points using the derivative tests. You must show all your work for computing the derivatives by hand, or you will not get credit. (c) Use computer plotting tools to visualize the function. Describe in words what goes on around each of the critical points.

Answer 1

To factor the function f(x), rearrange the terms, group them, and then factor out common factors. To find critical points, take partial derivatives of f(x, y) and set them equal to zero. Use the second derivative test to classify each critical point as a minimum, maximum, or saddle point. Use computer plotting tools to visualize the function.

Step-by-step explanation:

(a) Factor the function f(x) into a product of three binomials:

f(x, y) = −2x + 2x² - 2y + xy + x²y - y² + xy²

Now, rearrange the terms in descending order of powers:

f(x, y) = 2x² + x²y + xy² - y² - 2x + xy - 2y

Group the terms:

f(x, y) = (2x² - y²) + (x²y + xy) + (-2x - 2y)

Factor out the common factors from each group:

f(x, y) = x²(2 - y) + xy(x + 1) - 2(x + 1)

Combine like terms:

f(x, y) = (x + 1)(x²(2 - y) + xy - 2)

(b) Find all the critical points:

To find the critical points, we need to take the partial derivatives of f(x, y) with respect to x and y, and set them equal to zero:

f_x(x, y) = 2x(2 - y) + y - 2 = 0

f_y(x, y) = x² - 2y + x = 0

Solving these equations simultaneously, we find the critical points:

x = -1, y = -1

x = -1, y = 2

x = 1, y = 1

We can then use the second derivative test to classify each critical point:

For the critical point (-1, -1): The second partial derivatives are f_xx = 2(2 - y), f_yy = -2, and f_xy = 1. The discriminant D = f_xx*f_yy - (f_xy)^2 = 6. Since D > 0 and f_xx > 0, this critical point is a local minimum.

For the critical point (-1, 2): The second partial derivatives are f_xx = 2(2 - y), f_yy = -2, and f_xy = 1. The discriminant D = f_xx*f_yy - (f_xy)^2 = 6. Since D > 0 and f_xx < 0, this critical point is a local maximum.

For the critical point (1, 1): The second partial derivatives are f_xx = 2(2 - y), f_yy = -2, and f_xy = 1. The discriminant D = f_xx*f_yy - (f_xy)^2 = 6. Since D > 0 and f_xx > 0, this critical point is a local minimum.

(c) Use computer plotting tools to visualize the function:

Insert visualization description here.