1. **State the problem:** We are given the function $$f(x,y) = x^4 + y^4 + 4xy$$ and we want to analyze or work with it. 2. **Understand the function:** This is a function of two variables, $x$ and $y$, involving quartic terms $x^4$ and $y^4$ and a mixed term $4xy$. 3. **Find critical points (optional step):** To find critical points, we compute partial derivatives: $$\frac{\partial f}{\partial x} = 4x^3 + 4y$$ $$\frac{\partial f}{\partial y} = 4y^3 + 4x$$ 4. **Set partial derivatives to zero to find stationary points:** $$4x^3 + 4y = 0 \implies x^3 + y = 0$$ $$4y^3 + 4x = 0 \implies y^3 + x = 0$$ 5. **Solve the system:** From the first equation, $y = -x^3$. Substitute into the second: $$(-x^3)^3 + x = -x^9 + x = 0$$ $$x - x^9 = 0$$ $$x(1 - x^8) = 0$$ 6. **Solutions for $x$:** $$x = 0 \quad \text{or} \quad x^8 = 1 \implies x = \pm 1$$ 7. **Corresponding $y$ values:** - For $x=0$, $y = -0^3 = 0$ - For $x=1$, $y = -1^3 = -1$ - For $x=-1$, $y = -(-1)^3 = 1$ 8. **Critical points are:** $$(0,0), (1,-1), (-1,1)$$ 9. **Evaluate $f$ at critical points:** - $f(0,0) = 0 + 0 + 0 = 0$ - $f(1,-1) = 1 + 1 + 4(1)(-1) = 2 - 4 = -2$ - $f(-1,1) = 1 + 1 + 4(-1)(1) = 2 - 4 = -2$ 10. **Interpretation:** The function has critical points at these coordinates with values as above. The points $(1,-1)$ and $(-1,1)$ yield the minimum value $-2$ among these points. **Final answer:** The critical points of $$f(x,y) = x^4 + y^4 + 4xy$$ are $$(0,0), (1,-1), (-1,1)$$ with function values $$0, -2, -2$$ respectively.