classification options are: saddle point, local maximum, local minimum ### Educational Text for Critical Points Analysis**Task:**Use the level curves in the figure to predict the location of the critical points of the function $ f $ and determine whether $ f $ has a saddle point, a local maximum, or a local minimum at each critical point. Then, use the Second Derivatives Test to confirm your predictions. (Order your answers by their ordered pairs, from smallest to largest x-coordinate.)**Function:**\[ f(x, y) = 4 + x^3 + y^3 - 3xy \]**Critical Points to Identify:**1. $ (x, y) = \quad $ [Select Classification from options: Saddle Point, Local Maximum, Local Minimum]2. $ (x, y) = \quad $ [Select Classification from options: Saddle Point, Local Maximum, Local Minimum]**Graph Explanation:**The graph displays a set of level curves for the function $ f(x, y) = 4 + x^3 + y^3 - 3xy $ plotted on an $ xy $-coordinate plane. The level curves are labeled with numerical values near them, representing specific $ f(x, y) $ values.- **Axes:** The horizontal axis represents the $ x $-coordinate, while the vertical axis represents the $ y $-coordinate. Both axes are labeled neatly.- **Level Curves:** The curves are smooth and illustrate changes in the value of $ f $. The values increase from the lower left to the upper right, with notable regions around: - $ (1, 3.2) $ - $ (3.2, 0) $ - $ (3.2, 3.7) $ - $ (4.2, 1) $These curves indicate regions where $ f $ values are higher or lower, helping identify critical points visually. The curves suggest areas where the slope changes direction, likely indicating critical points (e.g., local maxima/minima, saddle points).**Instructions:**For each critical point you identify, select its classification using the dropdown menu provided next to each coordinate field. Once you've input your predictions, apply the Second Derivatives Test for confirmation.

Given a function, fx,y=4+x3+y3-3xy. From the level curves, In the neighbourhood of the point 1,1,…

Use the level curves in the figure to predict the location of the critical points of fand whether f has a saddle point or a local maximum or minimum at each critical point. Then use the Second Derivatives Test to confirm your predictions. (Order your answers by their ordered pairs, from smallest to largest x.) f(x, y) = 4 + x + y - 3xy (x, v) = (| Select Classification v (x, v) = (| Select Classification v 3.7 4.2 3.7 3.2 -1-

Use the level curves in the figure to predict the location of the critical points of fand whether f has a saddle point or a local maximum or minimum at each critical point. Then use the Second Derivatives Test to confirm your predictions. (Order your answers by their ordered pairs, from smallest to largest x.) f(x, y) = 4 + x + y - 3xy (x, v) = (| Select Classification v (x, v) = (| Select Classification v 3.7 4.2 3.7 3.2 -1-

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

classification options are: saddle point, local maximum, local minimum

### Educational Text for Critical Points Analysis

**Task:**
Use the level curves in the figure to predict the location of the critical points of the function $ f $ and determine whether $ f $ has a saddle point, a local maximum, or a local minimum at each critical point. Then, use the Second Derivatives Test to confirm your predictions. (Order your answers by their ordered pairs, from smallest to largest x-coordinate.)

**Function:**
\[ f(x, y) = 4 + x^3 + y^3 - 3xy \]

**Critical Points to Identify:**
1. $ (x, y) = \quad $ [Select Classification from options: Saddle Point, Local Maximum, Local Minimum]
2. $ (x, y) = \quad $ [Select Classification from options: Saddle Point, Local Maximum, Local Minimum]

**Graph Explanation:**
The graph displays a set of level curves for the function $ f(x, y) = 4 + x^3 + y^3 - 3xy $ plotted on an $ xy $-coordinate plane. The level curves are labeled with numerical values near them, representing specific $ f(x, y) $ values.

- **Axes:** The horizontal axis represents the $ x $-coordinate, while the vertical axis represents the $ y $-coordinate. Both axes are labeled neatly.
- **Level Curves:** The curves are smooth and illustrate changes in the value of $ f $. The values increase from the lower left to the upper right, with notable regions around:
- $ (1, 3.2) $
- $ (3.2, 0) $
- $ (3.2, 3.7) $
- $ (4.2, 1) $

These curves indicate regions where $ f $ values are higher or lower, helping identify critical points visually. The curves suggest areas where the slope changes direction, likely indicating critical points (e.g., local maxima/minima, saddle points).

**Instructions:**
For each critical point you identify, select its classification using the dropdown menu provided next to each coordinate field. Once you've input your predictions, apply the Second Derivatives Test for confirmation.

Formula Formula A function f ( x ) is also said to have attained a local minimum at x = a , if there exists a neighborhood ( a − δ , a + δ ) of a such that, f ( x ) > f ( a ) , ∀ x ∈ ( a − δ , a + δ ) , x ≠ a f ( x ) − f ( a ) > 0 , ∀ x ∈ ( a − δ , a + δ ) , x ≠ a In such a case f ( a ) is called the local minimum value of f ( x ) at x = a .

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