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.

Determine the behavior of the function at any of the critical points for which the Second Derivative Test is inconclusive. Select the correct choice below and, if necessary, fill in the answer box(es) within your choice.
A. Among these points, there are local minimum/minima at , and no local maxima or saddle points.
(Type an ordered pair. Use a comma to separate answers as needed.)
B. Among these points, there are saddle point(s) at , and no local maxima or minima.
(Type an ordered pair. Use a comma to separate answers as needed.)
C. Among these points, there are local maximum/maxima at , local minimum/minima at points. , and no saddle
(Type an ordered pair. Use a comma to separate answers as needed.)
D. Among these points, there are local maximum/maxima at , saddle point(s) at
(Type an ordered pair. Use a comma to separate answers as needed.)
, and no local minima.
E. Among these points, there are local maximum/maxima at at
, local minimum/minima at .
, and saddle point(s)
(Type an ordered pair. Use a comma to separate answers as needed.)

Ask by May Higgins. 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.

Determine the behavior of the function at any of the critical points for which the Second Derivative Test is inconclusive. Select the correct choice below and, if necessary, fill in the answer box(es) within your choice.
A. Among these points, there are local minimum/minima at , and no local maxima or saddle points.
(Type an ordered pair. Use a comma to separate answers as needed.)
B. Among these points, there are saddle point(s) at , and no local maxima or minima.
(Type an ordered pair. Use a comma to separate answers as needed.)
C. Among these points, there are local maximum/maxima at , local minimum/minima at points. , and no saddle
(Type an ordered pair. Use a comma to separate answers as needed.)
D. Among these points, there are local maximum/maxima at , saddle point(s) at
(Type an ordered pair. Use a comma to separate answers as needed.)
, and no local minima.
E. Among these points, there are local maximum/maxima at at
, local minimum/minima at .
, and saddle point(s)
(Type an ordered pair. Use a comma to separate answers as needed.)

Ask by May Higgins. in the United States

Mar 21,2025

UpStudy AI · Accepted Answer

We begin by finding the critical points.

1. Compute the first partial derivatives:
   $$
   f_x(x,y) = \frac{\partial}{\partial x} (-8x^2+5y^2-13) = -16x,\quad f_y(x,y) = \frac{\partial}{\partial y} (-8x^2+5y^2-13) = 10y.
   $$

2. Set the first partial derivatives equal to zero:
   $$
   -16x = 0 \quad \Rightarrow \quad x = 0,
   $$
   $$
   10y = 0 \quad \Rightarrow \quad y = 0.
   $$
   So, the only critical point is $$ (0,0) $$.

3. Next, we compute the second partial derivatives:
   $$
   f_{xx} = \frac{\partial^2 f}{\partial x^2} = -16,\quad f_{yy} = \frac{\partial^2 f}{\partial y^2} = 10,\quad f_{xy} = \frac{\partial^2 f}{\partial x \partial y} = 0.
   $$

4. Form the Hessian determinant:
   $$
   D = f_{xx} \cdot f_{yy} - (f_{xy})^2 = (-16)(10) - 0^2 = -160.
   $$
   Since $$ D < 0 $$, the Second Derivative Test indicates that the critical point $$ (0,0) $$ is a saddle point.

Thus, the function has a saddle point at $$ (0,0) $$ and no local minima or local maxima.

The correct choice is:

B. Among these points, there are saddle point(s) at $$ (0,0) $$, and no local maxima or minima.
The function has a saddle point at $$ (0,0) $$ and no local minima or maxima.

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