please describe each step clearly and if you could don't write it in cursive handwriting thank you! ### Question 7: Analysis of Extrema for a Continuous Function**Prompt:**The continuous function $ f(x,y) = x^3 + x + 2y $ has no critical points on the interior of the closed region D given below. Find the absolute extrema for $ f(x,y) $ over the closed region D.**Region D:**The region D is bounded by the parabola $ y = x^2 $ and the line $ y = 1 $.**Graph Description:**The graph shows the closed region D which is enclosed between the parabola $ y = x^2 $ and the horizontal line $ y = 1 $. The parabola opens upwards and intersects the horizontal line at two points, thereby forming a closed region with a shape similar to a horizontally oriented paraboloid cap.To find the absolute extrema of the function $ f(x,y) $ over the closed region D, we need to evaluate the function on the boundary of the region since there are no critical points inside D. The boundary consists of the curve $ y = x^2 $ and the lines where $ y = 1 $.### Steps:1. **Evaluate $ f(x,y) $ along the boundary:** - For $ y = 1 $: The equation of the line is $ y = 1 $. - For $ y = x^2 $: The equation of the parabola is $ y = x^2 $.2. **Substitute these boundary conditions into the function $ f(x,y) $ and find the maximum and minimum values by evaluating $ f(x,y) $ as a function of $ x $ along these boundaries.**3. **Check the endpoints where the boundary conditions intersect:** - Identify the intersection points of $ y = x^2 $ and $ y = 1 $. Finally, compare the values of $ f(x,y) $ at these points and determine the absolute maximum and minimum values over the closed region D.### Conceptual Understanding:- The absolute extrema (maximum or minimum values) of a continuous function over a closed region can occur at interior critical points or on the boundary of the region.- Since no critical points exist within the interior for the given function $ f(x,y) $, the extrema are found along the boundary.This problem is a practical application of the Extreme Value Theorem,

Answered: Image /qna-images/answer/b39a8632-44c1-46d3-8cb0-9edf1ef2ad41.jpg

7. The continuous function f (x,y) = x³ + x + 2y has no critical points on the interior of the closed region D given below. Find the absolute extrema for f (x, y) over the closed region D. y = 1 D y = x2

7. The continuous function f (x,y) = x³ + x + 2y has no critical points on the interior of the closed region D given below. Find the absolute extrema for f (x, y) over the closed region D. y = 1 D y = x2

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

please describe each step clearly and if you could don't write it in cursive handwriting

thank you!

### Question 7: Analysis of Extrema for a Continuous Function

**Prompt:**

The continuous function $ f(x,y) = x^3 + x + 2y $ has no critical points on the interior of the closed region D given below. Find the absolute extrema for $ f(x,y) $ over the closed region D.

**Region D:**

The region D is bounded by the parabola $ y = x^2 $ and the line $ y = 1 $.

**Graph Description:**

The graph shows the closed region D which is enclosed between the parabola $ y = x^2 $ and the horizontal line $ y = 1 $. The parabola opens upwards and intersects the horizontal line at two points, thereby forming a closed region with a shape similar to a horizontally oriented paraboloid cap.

To find the absolute extrema of the function $ f(x,y) $ over the closed region D, we need to evaluate the function on the boundary of the region since there are no critical points inside D. The boundary consists of the curve $ y = x^2 $ and the lines where $ y = 1 $.

### Steps:

1. **Evaluate $ f(x,y) $ along the boundary:**
- For $ y = 1 $: The equation of the line is $ y = 1 $.
- For $ y = x^2 $: The equation of the parabola is $ y = x^2 $.

2. **Substitute these boundary conditions into the function $ f(x,y) $ and find the maximum and minimum values by evaluating $ f(x,y) $ as a function of $ x $ along these boundaries.**

3. **Check the endpoints where the boundary conditions intersect:**
- Identify the intersection points of $ y = x^2 $ and $ y = 1 $.

Finally, compare the values of $ f(x,y) $ at these points and determine the absolute maximum and minimum values over the closed region D.

### Conceptual Understanding:

- The absolute extrema (maximum or minimum values) of a continuous function over a closed region can occur at interior critical points or on the boundary of the region.
- Since no critical points exist within the interior for the given function $ f(x,y) $, the extrema are found along the boundary.

This problem is a practical application of the Extreme Value Theorem,

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