**Graph Analysis: Identifying Critical Points**Consider the following graph and mark the critical points.**Function:**\[ x^2 e^{-\frac{x^2}{5}}, \, 0.2 \]**Graph Explanation:**- The graph shows the plot of the function $ x^2 e^{-\frac{x^2}{5}} $, which features exponential decay combined with a quadratic component.- The x-axis ranges from -2 to +2.- The y-axis ranges from 0 to 2.- The function exhibits a sinusoidal form with peaks and valleys, indicating the presence of multiple critical points (local maxima and minima).- There are noticeable maxima near $\pm 1.5$ and a noticeable minimum at the origin (0,0).**Steps to Identify Critical Points:**1. **Locate Maximum Points:** Find where the function peaks. - These are the points where the derivative of the function changes from positive to negative. 2. **Locate Minimum Points:** Find where the function dips lowest. - These are the points where the derivative of the function changes from negative to positive.**Graph Features:**- **x-axis (Horizontal):** Ranges from -2 to 2.- **y-axis (Vertical):** Ranges from 0 to 2.- **Gridlines:** Each grid cell marks an increment of 0.2 units in both the x and y directions.**How to Mark Critical Points:**1. **Find the derivative:** To determine critical points analytically, take the first derivative of the function and set it to zero.2. **Evaluate the second derivative:** To classify whether the critical points are minima or maxima, use the second derivative test.Use standard calculus techniques to find the precise coordinates where the function's slope is zero, and determine whether each point is a local minimum or maximum.Understanding and identifying these critical points on the graph will deepen your comprehension of the function's behavior and its real-world applications in analyzing data trends and optimizing solutions.

Critical points are calculated by the equating the slope of a function equal to 0. Slope equal to 0…

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Math

Calculus

Mark the critical points on the following graph. æ²e¯, 0.2 2+ 1.8 1.6 1.4 1.2- 0.8 0.6- 0.4

Calculus: Early Transcendentals

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

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Question

**Graph Analysis: Identifying Critical Points**

Consider the following graph and mark the critical points.

**Function:**
\[ x^2 e^{-\frac{x^2}{5}}, \, 0.2 \]

**Graph Explanation:**

- The graph shows the plot of the function $ x^2 e^{-\frac{x^2}{5}} $, which features exponential decay combined with a quadratic component.
- The x-axis ranges from -2 to +2.
- The y-axis ranges from 0 to 2.
- The function exhibits a sinusoidal form with peaks and valleys, indicating the presence of multiple critical points (local maxima and minima).
- There are noticeable maxima near $\pm 1.5$ and a noticeable minimum at the origin (0,0).

**Steps to Identify Critical Points:**

1. **Locate Maximum Points:** Find where the function peaks.
- These are the points where the derivative of the function changes from positive to negative.

2. **Locate Minimum Points:** Find where the function dips lowest.
- These are the points where the derivative of the function changes from negative to positive.

**Graph Features:**

- **x-axis (Horizontal):** Ranges from -2 to 2.
- **y-axis (Vertical):** Ranges from 0 to 2.
- **Gridlines:** Each grid cell marks an increment of 0.2 units in both the x and y directions.

**How to Mark Critical Points:**

1. **Find the derivative:** To determine critical points analytically, take the first derivative of the function and set it to zero.
2. **Evaluate the second derivative:** To classify whether the critical points are minima or maxima, use the second derivative test.

Use standard calculus techniques to find the precise coordinates where the function's slope is zero, and determine whether each point is a local minimum or maximum.

Understanding and identifying these critical points on the graph will deepen your comprehension of the function's behavior and its real-world applications in analyzing data trends and optimizing solutions.

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