To find the critical points of the function $f(x, y) = xy - \frac{1}{4}x^4 - 2y^2$, we need to compute the partial derivatives and set them equal to zero. Let's proceed step by step:

1. Compute the partial derivatives:

   • $f_x(x, y) = \frac{\partial}{\partial x} \left(xy - \frac{1}{4}x^4 - 2y^2\right) = y - x^3$
   • $f_y(x, y) = \frac{\partial}{\partial y} \left(xy - \frac{1}{4}x^4 - 2y^2\right) = x - 4y$

2. Set the partial derivatives equal to zero:

   • $f_x(x, y) = y - x^3 = 0$
   • $f_y(x, y) = x - 4y = 0$

   Solving these equations will give the critical points.

3. Solve the system of equations:

   • From $f_y(x, y) = 0$: $x = 4y$
   • Substitute $x = 4y$ into $f_x(x, y) = 0$: $y - (4y)^3 = 0$ $y - 64y^3 = 0$ $y(1 - 64y^2) = 0$

   This gives:

   • $y = 0$
   • $1 - 64y^2 = 0$ → $y^2 = \frac{1}{64}$ → $y = \pm\frac{1}{8}$

4. Find corresponding $x$ values:

   • For $y = 0$: $x = 4(0) = 0$, so $(x, y) = (0, 0)$
   • For $y = \frac{1}{8}$: $x = 4\left(\frac{1}{8}\right) = \frac{1}{2}$, so $(x, y) = \left(\frac{1}{2}, \frac{1}{8}\right)$
   • For $y = -\frac{1}{8}$: $x = 4\left(-\frac{1}{8}\right) = -\frac{1}{2}$, so $(x, y) = \left(-\frac{1}{2}, -\frac{1}{8}\right)$

5. Conclusion: The critical points are:

   • $(0, 0)$
   • $\left(\frac{1}{2}, \frac{1}{8}\right)$
   • $\left(-\frac{1}{2}, -\frac{1}{8}\right)$

   Hence, there are 3 critical points.

Would you like to analyze the nature of these critical points (e.g., maximum, minimum, or saddle)? Let me know if you'd like a detailed explanation or further questions.

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Related Questions:

1. How can you classify the critical points of this function?
2. What are the second partial derivatives of $f(x, y)$, and how are they used to classify critical points?
3. How does the presence of $x^4$ and $y^2$ terms affect the critical points geometrically?
4. Could you plot $f(x, y)$ to visualize the critical points?
5. What happens if an additional term (e.g., $+z^2$) is added to the function?

Tip:

Always confirm critical point classification by using the second derivative test or analyzing the Hessian matrix.