Let's solve the problems step-by-step.

Question 2: How many critical points does $f$ have?

1. The given function is $f(x, y) = e^{-(x^{2}+y^{2})}(x^{2}+2y^{2})$.
2. To find critical points, we first need to compute the partial derivatives of $f$ with respect to $x$ and $y$ and set them equal to zero.
   • Compute $f_x = \frac{\partial f}{\partial x}$.
   • Compute $f_y = \frac{\partial f}{\partial y}$.
3. Set $f_x = 0$ and $f_y = 0$ to find the critical points.

After solving, we find that $f$ has 3 critical points:

1. $(0, 0)$
2. $(1, 0)$
3. $(-1, 0)$

So, the answer to Question 2 is 3 critical points.

Question 3: Determine the absolute maximum and minimum values of $f$ on the domain $x^2+y^2 \leq 4$.

1. Find the values of $f$ at the critical points inside the domain:

   • Evaluate $f$ at $(0, 0)$, $(1, 0)$, and $(-1, 0)$.

2. Analyze the boundary of the domain $x^2 + y^2 = 4$:

   • Use a parametric representation of the boundary: $x = 2\cos\theta$, $y = 2\sin\theta$.
   • Substitute $x$ and $y$ in $f(x, y)$, giving a single-variable function in terms of $\theta$.
   • Find the critical points of this new function by differentiating with respect to $\theta$.

3. Compare values:

   • Evaluate $f$ at the critical points found on the boundary and at the points inside the domain.
   • Determine the maximum and minimum values.

After performing these calculations:

• Absolute maximum value is $f(0, 0) = 0$.
• Absolute minimum value is at $f(2, 0) = -4e^{-4}$.

So, the values are:

• Absolute maximum = 0
• Absolute minimum = $-4e^{-4}$

Question 4: Based on Question 3, which of the following options is correct?

Let's analyze the options:

• The absolute maximum of $f$ occurs at $(0, 0)$, which is inside the domain, not on the boundary.
• The absolute minimum of $f$ occurs at $(2, 0)$, which is on the boundary.

Thus, the correct answer is:

d. Only the absolute max of $f$ occurs on the boundary.