**Mathematical Analysis: Sketching a Graph****Problem Statement:**Suppose that $ f(3) = 2 $, $ f'(3) = \frac{1}{2} $, and $ f'(x) > 0 $ and $ f''(x) < 0 $ for all values of $ x $ in $ (-\infty, \infty) $. Sketch a possible graph of $ f(x) $.**Instructions and Explanation:**1. **Given Information:** - The function value at $ x = 3 $ is $ f(3) = 2 $. - The derivative at $ x = 3 $ is $ f'(3) = \frac{1}{2} $, indicating the slope of the tangent line at this point. - The derivative $ f'(x) > 0 $ for all $ x $, meaning the function is increasing everywhere. - The second derivative $ f''(x) < 0 $ for all $ x $, indicating the function is concave down everywhere.2. **Graph Characteristics:** - The graph passes through the point $ (3, 2) $. - Since $ f'(x) > 0 $, the graph is continuously increasing from left to right. - Since $ f''(x) < 0 $, the graph is concave down, resembling the shape of an upside-down bowl.3. **Sketching the Graph:** - Start the graph at the given point $ (3, 2) $. - Ensure that the curve is increasing throughout its domain. - Make sure that the graph bends downwards, maintaining concavity.By following these steps, you will have a graph that accurately represents the mathematical descriptions provided.

Answered: Image /qna-images/answer/58403300-9018-4e93-ba55-4f3f8358c575.jpg

Answered: Suppose that f(3) = 2, f'(3) = 5, and…

Suppose that f(3) = 2, f'(3) = 5, and f' (x) > 0 and f"(x) < 0 for all values of x in (-0, 00). Sketch a possible graph of f(x)

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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Concept explainers

Transformation of Graphs

The word ‘transformation’ means modification. Transformation of the graph of a function is a process by which we modify or change the original graph and make a new graph.

Exponential Functions

The exponential function is a type of mathematical function which is used in real-world contexts. It helps to find out the exponential decay model or exponential growth model, in mathematical models. In this topic, we will understand descriptive rules, concepts, structures, graphs, interpreter series, work formulas, and examples of functions involving exponents.

Question

**Mathematical Analysis: Sketching a Graph**

**Problem Statement:**

Suppose that $ f(3) = 2 $, $ f'(3) = \frac{1}{2} $, and $ f'(x) > 0 $ and $ f''(x) < 0 $ for all values of $ x $ in $ (-\infty, \infty) $. Sketch a possible graph of $ f(x) $.

**Instructions and Explanation:**

1. **Given Information:**
- The function value at $ x = 3 $ is $ f(3) = 2 $.
- The derivative at $ x = 3 $ is $ f'(3) = \frac{1}{2} $, indicating the slope of the tangent line at this point.
- The derivative $ f'(x) > 0 $ for all $ x $, meaning the function is increasing everywhere.
- The second derivative $ f''(x) < 0 $ for all $ x $, indicating the function is concave down everywhere.

2. **Graph Characteristics:**
- The graph passes through the point $ (3, 2) $.
- Since $ f'(x) > 0 $, the graph is continuously increasing from left to right.
- Since $ f''(x) < 0 $, the graph is concave down, resembling the shape of an upside-down bowl.

3. **Sketching the Graph:**
- Start the graph at the given point $ (3, 2) $.
- Ensure that the curve is increasing throughout its domain.
- Make sure that the graph bends downwards, maintaining concavity.

By following these steps, you will have a graph that accurately represents the mathematical descriptions provided.

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