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 Martin Delgado. 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 Martin Delgado. in the United States

Mar 21,2025

UpStudy AI · Accepted Answer

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

1. **Find the critical points:**

Compute the first partial derivatives:
   $$
   f_x = \frac{\partial f}{\partial x} = -16x, \quad f_y = \frac{\partial f}{\partial y} = 10y.
   $$
   
   Set them equal to zero:
   $$
   -16x = 0 \quad \Rightarrow \quad x = 0,
   $$
   $$
   10y = 0 \quad \Rightarrow \quad y = 0.
   $$
   
   Thus, the only critical point is $$(0, 0)$$.

2. **Apply the Second Derivative Test:**

Compute the second partial derivatives:
   $$
   f_{xx} = \frac{\partial^2 f}{\partial x^2} = -16, \quad f_{yy} = \frac{\partial^2 f}{\partial y^2} = 10, \quad f_{xy} = \frac{\partial^2 f}{\partial x \partial y} = 0.
   $$
   
   Calculate the determinant of the Hessian matrix:
   $$
   D = f_{xx} f_{yy} - \left(f_{xy}\right)^2 = (-16)(10) - 0 = -160.
   $$
   
   Since $$ D < 0 $$, the critical point $$(0, 0)$$ is a saddle point.

The only critical point is $$(0, 0)$$, and it corresponds to a saddle point.
The only critical point is $$(0, 0)$$, and it is a saddle point.

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