**Graph Description:**The graph provided is of a function $ f(x) $. It shows a curve with certain key points and characteristics:- It has an upward direction at both ends.- There is a local maximum point around $ x = -1.4 $ with coordinates approximately $(-1.4, 39.6)$.- The graph crosses the origin at point $(0, 0)$.- There is a local minimum point around $ x = 1.4 $ with coordinates approximately $(1.4, -39.6)$.**Questions:**a) **Where is $ f'(x) > 0?** To determine where the derivative \( f'(x) $ is positive, look for intervals where the graph is increasing. This occurs between the local minimum and maximum points.b) **Where does $ f $ have a critical number?** Critical numbers occur where $ f'(x) = 0 $ or where $ f'(x) $ is undefined. These correspond to the local maximum and minimum points.c) **Where is $ f $ concave down?** Concave down regions are where the graph appears to curve downwards. This typically occurs between inflection points, where the concavity changes from up to down.d) **State all inflection points.** Inflection points are where the graph changes concavity. These are found between regions of convexity. The precise coordinates are not directly labeled on the graph, so they must be inferred by analysis of the graph's curvature.

Answered: Image /qna-images/answer/69c4338b-7651-4fff-8c3e-f659b41b3b66.jpg

1. Consider the graph below. 70 + a) Where is f'(x) > 0? 60 50 + 40 - (--1.4-39.6) 30 b) Where does f have a critical number? 20 10 (0, 0) -2 c) Where is f concave down? -3 -10 -20 + -30 + (-1.4 --39.6) -40 d) State all inflection points. -50 -60 2. -1

1. Consider the graph below. 70 + a) Where is f'(x) > 0? 60 50 + 40 - (--1.4-39.6) 30 b) Where does f have a critical number? 20 10 (0, 0) -2 c) Where is f concave down? -3 -10 -20 + -30 + (-1.4 --39.6) -40 d) State all inflection points. -50 -60 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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Numerical Integration

In a mathematical investigation, numerical integration comprises a wide group of calculations for computing the mathematical estimation of definite integral, and likewise, the term is moreover in some cases used to depict the numerical solution of differential equations. The most important aspect of this theory is error analysis.

Question

**Graph Description:**

The graph provided is of a function $ f(x) $. It shows a curve with certain key points and characteristics:

- It has an upward direction at both ends.
- There is a local maximum point around $ x = -1.4 $ with coordinates approximately $(-1.4, 39.6)$.
- The graph crosses the origin at point $(0, 0)$.
- There is a local minimum point around $ x = 1.4 $ with coordinates approximately $(1.4, -39.6)$.

**Questions:**

a) **Where is $ f'(x) > 0?**

To determine where the derivative \( f'(x) $ is positive, look for intervals where the graph is increasing. This occurs between the local minimum and maximum points.

b) **Where does $ f $ have a critical number?**

Critical numbers occur where $ f'(x) = 0 $ or where $ f'(x) $ is undefined. These correspond to the local maximum and minimum points.

c) **Where is $ f $ concave down?**

Concave down regions are where the graph appears to curve downwards. This typically occurs between inflection points, where the concavity changes from up to down.

d) **State all inflection points.**

Inflection points are where the graph changes concavity. These are found between regions of convexity. The precise coordinates are not directly labeled on the graph, so they must be inferred by analysis of the graph's curvature.

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