Final answer:
The critical points of the system are found by setting the gradient of the function to zero. After calculating the partial derivatives and setting them to zero, two critical points, (1, 0) and (1, 1), are found.
Step-by-step explanation:
Finding Critical Points of a Multivariable Function
To find the critical points of the function f(x1, x2) = (x1^2 - x2)^2 - (x2 - 1)^2, we need to find the points where the gradient of f is zero. The gradient of f is a vector of partial derivatives of f with respect to x1 and x2. Thus, we first calculate the partial derivatives:
∂f/∂x1 = 2(x1^2 - x2)(2x1) = 4x1(x1^2 - x2)
∂f/∂x2 = 2(x1^2 - x2)(-1) - 2(x2 - 1) = -2(x1^2 - x2) - 2x2 + 2
We then set these partial derivatives to zero to solve for the points (x1, x2).
- 4x1(x1^2 - x2) = 0
- -2(x1^2 - x2) - 2x2 + 2 = 0
From the first equation, we have two possibilities: either x1 = 0 or x1^2 = x2. If x1 = 0, substituting into the second equation gives us a critical point at (0, 1). If x1^2 = x2, we substitute x2 into the second equation to find another critical point at (1, 1).
Therefore, the critical points of the system are (0, 1) and (1, 1), making option 'c' incorrect, and the correct answer is 'd': (1, 0) and (1, 1).