Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter3: Functions And Graphs
Section3.5: Graphs Of Functions
Problem 36E
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![### Finding the Critical Point of the Function
To determine the critical point of the given function \( f(x, y) \), follow the steps outlined below:
#### Given Function:
\[ f(x, y) = -1 + 7x - 6x^2 - 7y + 5y^2 \]
#### Task:
1. Find the critical point of the function \( f(x, y) \).
2. Identify the type of critical point.
#### Solution:
1. **Calculating Partial Derivatives:**
To find the critical points, we need to calculate the first partial derivatives of \( f(x, y) \) with respect to \( x \) and \( y \), and set them equal to 0.
- \[ f_x = \frac{\partial f}{\partial x} \]
- \[ f_y = \frac{\partial f}{\partial y} \]
2. **Solving for \( x \) and \( y \):**
Solve the system of equations obtained from the partial derivatives to find the values of \( x \) and \( y \) at the critical point.
3. **Determine the Nature of the Critical Point:**
Use the second partial derivative test to classify the critical point as local minimum, local maximum, or saddle.
#### Input Form:
- **Text Box**: Enter the critical point coordinates \((x, y)\).
- **Dropdown Menu**: Select the nature of the critical point.
Options available in the dropdown:
- Saddle
- Local Maximum
- Local Minimum
In this particular case, the critical point is identified as a **Saddle**.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3aff424b-0e96-4e90-b1d6-dc40303839e0%2Ff3a11eb6-9ee3-41fa-84ae-412dc648c7cf%2Fi17wxjl_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Finding the Critical Point of the Function
To determine the critical point of the given function \( f(x, y) \), follow the steps outlined below:
#### Given Function:
\[ f(x, y) = -1 + 7x - 6x^2 - 7y + 5y^2 \]
#### Task:
1. Find the critical point of the function \( f(x, y) \).
2. Identify the type of critical point.
#### Solution:
1. **Calculating Partial Derivatives:**
To find the critical points, we need to calculate the first partial derivatives of \( f(x, y) \) with respect to \( x \) and \( y \), and set them equal to 0.
- \[ f_x = \frac{\partial f}{\partial x} \]
- \[ f_y = \frac{\partial f}{\partial y} \]
2. **Solving for \( x \) and \( y \):**
Solve the system of equations obtained from the partial derivatives to find the values of \( x \) and \( y \) at the critical point.
3. **Determine the Nature of the Critical Point:**
Use the second partial derivative test to classify the critical point as local minimum, local maximum, or saddle.
#### Input Form:
- **Text Box**: Enter the critical point coordinates \((x, y)\).
- **Dropdown Menu**: Select the nature of the critical point.
Options available in the dropdown:
- Saddle
- Local Maximum
- Local Minimum
In this particular case, the critical point is identified as a **Saddle**.
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