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.)
Use the Second Derivative Test to find the local maxima. Select the correct choice below and fill in any answer boxes
within your choice.
A. The test shows that there is/are local maxima at
(Type an ordered pair. Use a comma to separate answers as needed.)
B. The test does not reveal any local maxima and there are no critical points for which the test is inconclusive, so
there are no local maxima.
c. The test does not reveal any local maxima, but there is at least one critical point for which the test is
inconclusive.

Ask by Ford Kelley. 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.)
Use the Second Derivative Test to find the local maxima. Select the correct choice below and fill in any answer boxes
within your choice.
A. The test shows that there is/are local maxima at
(Type an ordered pair. Use a comma to separate answers as needed.)
B. The test does not reveal any local maxima and there are no critical points for which the test is inconclusive, so
there are no local maxima.
c. The test does not reveal any local maxima, but there is at least one critical point for which the test is
inconclusive.

Ask by Ford Kelley. in the United States

Mar 21,2025

UpStudy AI · Accepted Answer

To find the critical points of the function $$ f(x, y) = x^{4} + y^{4} + 4x + 4y + 8 $$, we need to follow these steps:

1. **Find the first derivatives** $$ f_x $$ and $$ f_y $$.
2. **Set the first derivatives equal to zero** to find the critical points.
3. **Find the second derivatives** $$ f_{xx} $$, $$ f_{yy} $$, and $$ f_{xy} $$.
4. **Use the Second Derivative Test** to classify the critical points.

### Step 1: Find the first derivatives

The first derivatives are:
$$
f_x = \frac{\partial f}{\partial x} = 4x^{3} + 4
$$
$$
f_y = \frac{\partial f}{\partial y} = 4y^{3} + 4
$$

### Step 2: Set the first derivatives equal to zero

Setting $$ f_x = 0 $$ and $$ f_y = 0 $$:
$$
4x^{3} + 4 = 0 \implies x^{3} = -1 \implies x = -1
$$
$$
4y^{3} + 4 = 0 \implies y^{3} = -1 \implies y = -1
$$

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

### Step 3: Find the second derivatives

The second derivatives are:
$$
f_{xx} = \frac{\partial^2 f}{\partial x^2} = 12x^{2}
$$
$$
f_{yy} = \frac{\partial^2 f}{\partial y^2} = 12y^{2}
$$
$$
f_{xy} = \frac{\partial^2 f}{\partial x \partial y} = 0
$$

### Step 4: Evaluate the second derivatives at the critical point

Now we evaluate the second derivatives at the critical point $$ (-1, -1) $$:
$$
f_{xx}(-1, -1) = 12(-1)^{2} = 12
$$
$$
f_{yy}(-1, -1) = 12(-1)^{2} = 12
$$
$$
f_{xy}(-1, -1) = 0
$$

### Step 5: Apply the Second Derivative Test

The determinant $$ D $$ is given by:
$$
D = f_{xx} f_{yy} - (f_{xy})^{2} = 12 \cdot 12 - 0^{2} = 144
$$

Since $$ D > 0 $$ and $$ f_{xx} > 0 $$, the critical point $$ (-1, -1) $$ is a local minimum.

### Conclusion

- The critical point is $$ (-1, -1) $$.
- There are local minima at $$ (-1, -1) $$.

Thus, the answer is:

**Critical Points:** $$ (-1, -1) $$

**Local Maxima:** 
A. The test shows that there is/are local maxima at $$ (-1, -1) $$.
**Critical Points:** $$ (-1, -1) $$ **Local Maxima:** A. The test shows that there is a local minimum at $$ (-1, -1) $$.

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Solución

Step 1: Find the first derivatives

Step 2: Set the first derivatives equal to zero

Step 3: Find the second derivatives

Step 4: Evaluate the second derivatives at the critical point

Step 5: Apply the Second Derivative Test

Conclusion

Beyond the Answer

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