Final answer:
To find the critical points of the function f(x, y) = (1 - x^2/2)(y^2 + 2x^2 + 1), set the partial derivatives with respect to x and y equal to zero. The critical points are (0, 0) and (2, 0).
Step-by-step explanation:
The critical points of the function f(x, y) = (1 - x^2/2)(y^2 + 2x^2 + 1) can be found by taking the partial derivatives with respect to x and y and setting them equal to zero.
To find the critical points of f(x, y), we take the derivative of f(x, y) with respect to x and y, setting both derivatives equal to zero:
d/dx((1 - x^2/2)(y^2 + 2x^2 + 1)) = -x(y^2 + 2x^2 + 1) = 0
d/dy((1 - x^2/2)(y^2 + 2x^2 + 1)) = 2y(1 - x^2/2) = 0
Solving these equations simultaneously, we find that the critical points occur at (x, y) = (0, 0) and (x, y) = (2, 0).