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.

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

Ask by Norris Bond. 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.

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

Ask by Norris Bond. in the United States

Mar 21,2025

UpStudy AI · Accepted Answer

**Step 1. Find the critical points**

The function is  
$$
f(x,y) = -8x^2 + 5y^2 - 13.
$$

Compute the partial derivatives:

$$
f_x(x,y) = \frac{\partial f}{\partial x} = -16x,
$$
$$
f_y(x,y) = \frac{\partial f}{\partial y} = 10y.
$$

Set the partial derivatives equal to zero:

$$
-16x = 0 \quad \Rightarrow \quad x = 0,
$$
$$
10y = 0 \quad \Rightarrow \quad y = 0.
$$

Thus, the only critical point is $$(0,0)$$.

---

**Step 2. Use the Second Derivative Test**

Find the second-order partial derivatives:

$$
f_{xx}(x,y) = -16, \quad f_{yy}(x,y) = 10, \quad f_{xy}(x,y) = 0.
$$

Compute the determinant $$D$$ of the Hessian matrix at the critical point $$(0,0)$$:

$$
D = f_{xx}(0,0)f_{yy}(0,0) - \left(f_{xy}(0,0)\right)^2 = (-16)(10) - 0^2 = -160.
$$

Since $$D < 0$$, the Second Derivative Test indicates that the critical point $$(0,0)$$ is a saddle point.

---

**Final Answers**

- **Local Minima:**  
  **B.** The test does not reveal any local minima and there are no critical points for which the test is inconclusive, so there are no local minima.

- **Saddle Points:**  
  **A.** There is/are saddle point(s) at $$(0,0)$$.
**Local Minima:** B. There are no local minima. **Saddle Points:** A. There is a saddle point at (0,0).

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