Find the critical points of the following function. Use the Second Derivative Test to determine (if possible) whether each
critical point corresponds to a local maximum, local minimum, or saddle point. If the Second Derivative Test
is inconclusive, determine the behavior of the function at the critical points.

What are the critical points?
(Type an ordered pair. Use a comma to separate answers as needed.)
?

Ask by Mullins Wright. in the United States

Mar 21,2025

Question

Find the critical points of the following function. Use the Second Derivative Test to determine (if possible) whether each
critical point corresponds to a local maximum, local minimum, or saddle point. If the Second Derivative Test
is inconclusive, determine the behavior of the function at the critical points.

What are the critical points?
(Type an ordered pair. Use a comma to separate answers as needed.)
?

Ask by Mullins Wright. in the United States

Mar 21,2025

UpStudy AI · Accepted Answer

**Step 1. Find the Critical Points**

For the function
$$
f(x, y)=x^{4}+y^{4}+4x+4y+8,
$$
we first compute the partial derivatives.

The partial derivative with respect to $$ x $$ is:
$$
f_x(x,y)=\frac{\partial f}{\partial x}=4x^3+4.
$$
Setting $$ f_x(x,y) = 0 $$:
$$
4x^3+4=0 \quad \Rightarrow \quad x^3=-1 \quad \Rightarrow \quad x=-1.
$$

Similarly, the partial derivative with respect to $$ y $$ is:
$$
f_y(x,y)=\frac{\partial f}{\partial y}=4y^3+4.
$$
Setting $$ f_y(x,y) = 0 $$:
$$
4y^3+4=0 \quad \Rightarrow \quad y^3=-1 \quad \Rightarrow \quad y=-1.
$$

Thus, the only critical point is:
$$
(-1, -1).
$$

**Step 2. Second Derivative Test**

Next, compute the second partial derivatives.

$$
f_{xx}(x,y)=\frac{\partial^2 f}{\partial x^2}=12x^2,
$$
$$
f_{yy}(x,y)=\frac{\partial^2 f}{\partial y^2}=12y^2,
$$
$$
f_{xy}(x,y)=\frac{\partial^2 f}{\partial x \partial y}=0.
$$

At the critical point $$(-1, -1)$$, we have:
$$
f_{xx}(-1,-1)=12(-1)^2=12,
$$
$$
f_{yy}(-1,-1)=12(-1)^2=12,
$$
$$
f_{xy}(-1,-1)=0.
$$

Calculate the determinant of the Hessian matrix at $$(-1, -1)$$:
$$
D = f_{xx}(-1,-1) \cdot f_{yy}(-1,-1) - \left[f_{xy}(-1,-1)\right]^2 = 12 \times 12 - 0^2 = 144.
$$

Since $$ D > 0 $$ and $$ f_{xx}(-1,-1)>0 $$, the Second Derivative Test tells us that $$(-1, -1)$$ is a local minimum.

**Final Answer**

$$
(-1, -1)
$$
The critical point is $$(-1, -1)$$, which is a local minimum.

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