Question

Find the critical points for the function

f(x, y) = x²+ y³- 9x² -12y- 4
and classify each as a local maximum, local minimum, saddle point, or none of these.
critical points:_____
classifications:_____

Answer 1

The critical points for the function f(x, y) = x² + y³ - 9x² - 12y - 4 are (3, -2) and (0, -4). The classification of each critical point is as follows:

(3, -2) is a local minimum.

(0, -4) is a saddle point.

Step-by-step explanation:

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

Calculate ∂f/∂x and ∂f/∂y:

∂f/∂x = 2x - 18x

∂f/∂y = 3y² - 12

Set the partial derivatives equal to zero and solve for x and y:

2x - 18x = 0 → x = 0

3y² - 12 = 0 → y = ±2

So, we have critical points at (0, 2) and (0, -2).

Evaluate the second-order partial derivatives:

∂²f/∂x² = 2 - 18 = -16

∂²f/∂y² = 6y

Use the second-order partial derivatives in the Second Derivative Test to classify the critical points:

At (0, -2), ∂²f/∂x² is negative, so it's a local maximum.

At (0, 2), ∂²f/∂x² is negative, so it's a local maximum.

Therefore, the critical points are (0, 2) and (0, -2), and they are both local maxima.