A function is continuous on the closed interval \([-3, 3]\) such that \(f(-3) = 4\) and \(f(3) = 1\). The functions \(f'(x)\) and \(f''(x)\) have the properties given in the table below.| \(x\) | \(f'(x)\) | \(f''(x)\) ||-------------------|-------------|-------------|| \(-3 < x < -1\) | Positive | Positive || \(x = -1\) | Fails to exist | Fails to exist || \(-1 < x < 1\) | Negative | Positive || \(x = 1\) | Zero | Zero || \(1 < x < 3\) | Negative | Negative |Questions:a. What are the x-coordinates of all absolute maximum and minimum points of \(f\) on the interval \([-3, 3]\)? Justify your answer.b. What are the x-coordinates of all points of inflection on the interval \([-3, 3]\)? Justify your answer.c. On the axes provided, sketch a graph that satisfies the given properties of \(f\).**Graph Sketching:**- The graph should show increasing behavior on \((-3, -1)\) because \(f'(x) > 0\).- \(f(x)\) has a sudden change at \(x = -1\) where \(f'(x)\) fails to exist.- The graph should show decreasing behavior on \((-1, 1)\) since \(f'(x) < 0\), but with positive concavity (\(f''(x) > 0\)).- At \(x = 1\), \(f'(x) = 0\) and \(f''(x) = 0\), indicating a potential inflection point.- The graph continues to decrease on \((1, 3)\) with negative concavity (\(f''(x) < 0\)).These insights can be used to help sketch the function \(f(x)\) accurately.

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A function is continuous on the closed interval [-3,3] such that f(-3) = 4 and f(3) = 1. The functions f'(x) and f"(x) have the properties given in the table below. f'(x) f"(x) -3

Algebra and Trigonometry (6th Edition)

6th Edition

ISBN:9780134463216

Author:Robert F. Blitzer

Publisher:Robert F. Blitzer

ChapterP: Prerequisites: Fundamental Concepts Of Algebra

Section: Chapter Questions

Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...

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Question

A function is continuous on the closed interval \([-3, 3]\) such that \(f(-3) = 4\) and \(f(3) = 1\). The functions \(f'(x)\) and \(f''(x)\) have the properties given in the table below.

| \(x\) | \(f'(x)\) | \(f''(x)\) |
|-------------------|-------------|-------------|
| \(-3 < x < -1\) | Positive | Positive |
| \(x = -1\) | Fails to exist | Fails to exist |
| \(-1 < x < 1\) | Negative | Positive |
| \(x = 1\) | Zero | Zero |
| \(1 < x < 3\) | Negative | Negative |

Questions:
a. What are the x-coordinates of all absolute maximum and minimum points of \(f\) on the interval \([-3, 3]\)? Justify your answer.

b. What are the x-coordinates of all points of inflection on the interval \([-3, 3]\)? Justify your answer.

c. On the axes provided, sketch a graph that satisfies the given properties of \(f\).

**Graph Sketching:**

- The graph should show increasing behavior on \((-3, -1)\) because \(f'(x) > 0\).
- \(f(x)\) has a sudden change at \(x = -1\) where \(f'(x)\) fails to exist.
- The graph should show decreasing behavior on \((-1, 1)\) since \(f'(x) < 0\), but with positive concavity (\(f''(x) > 0\)).
- At \(x = 1\), \(f'(x) = 0\) and \(f''(x) = 0\), indicating a potential inflection point.
- The graph continues to decrease on \((1, 3)\) with negative concavity (\(f''(x) < 0\)).

These insights can be used to help sketch the function \(f(x)\) accurately.

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