The image contains four separate graphs, each depicting a polynomial function. Each graph is labeled with a checkbox on the left side:### Graph A:- **Description**: The graph shows a curve with one local maximum and one local minimum. It begins in the top left quadrant, dips down crossing the x-axis, rises to a local maximum just above the x-axis, and then declines steeply into the bottom right quadrant.- **Key Features**: - The curve has a local minimum around x = -2. - It has a local maximum near x = 1. - Intercepts the y-axis above the origin and the x-axis in two places.### Graph B:- **Description**: This graph has one inflection point and an overall positive slope. The curve rises from the bottom left to the top right.- **Key Features**: - Shows a point of inflection near x = 0. - A local minimum is visible before the inflection point. - The graph passes through the origin.### Graph C:- **Description**: The graph displays two local maxima and two local minima. It starts in the top left, forms a peak, then a trough, followed by another peak, and ends descending on the right.- **Key Features**: - A local maximum is present at about x = -3. - A local minimum occurs around x = -1. - Another peak is seen at x = 2.### Graph D:- **Description**: This graph has one local minimum and two points where it approaches a steep incline. The curve rises from the bottom left, dips downward, and then rises steeply.- **Key Features**: - Local minimum present near x = 1. - The curve increases again, indicating a strong upward slope beyond the minimum. - It crosses the y-axis above the origin.These graphs illustrate various behaviors of polynomial functions, such as changes in direction, local extrema, and inflection points, which are crucial for understanding calculus concepts like differentiation and integration. **Analyzing Functions Using Derivatives 5.9**The derivative of a certain continuous and differentiable function is given by:\[ y' = 3x^2 - 12x + 9 \]Analyze the function and create a sketch of the original if the only zeros are at $ x = 0 $ and $ x = 3 $.*This content is part of the MATH 165, Fall 2020, Online Midterm 1.*Note: There are no graphs or diagrams provided in the image.

Answered: The derivative of a certain continuous…

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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The image contains four separate graphs, each depicting a polynomial function. Each graph is labeled with a checkbox on the left side:

### Graph A:
- **Description**: The graph shows a curve with one local maximum and one local minimum. It begins in the top left quadrant, dips down crossing the x-axis, rises to a local maximum just above the x-axis, and then declines steeply into the bottom right quadrant.
- **Key Features**:
- The curve has a local minimum around x = -2.
- It has a local maximum near x = 1.
- Intercepts the y-axis above the origin and the x-axis in two places.

### Graph B:
- **Description**: This graph has one inflection point and an overall positive slope. The curve rises from the bottom left to the top right.
- **Key Features**:
- Shows a point of inflection near x = 0.
- A local minimum is visible before the inflection point.
- The graph passes through the origin.

### Graph C:
- **Description**: The graph displays two local maxima and two local minima. It starts in the top left, forms a peak, then a trough, followed by another peak, and ends descending on the right.
- **Key Features**:
- A local maximum is present at about x = -3.
- A local minimum occurs around x = -1.
- Another peak is seen at x = 2.

### Graph D:
- **Description**: This graph has one local minimum and two points where it approaches a steep incline. The curve rises from the bottom left, dips downward, and then rises steeply.
- **Key Features**:
- Local minimum present near x = 1.
- The curve increases again, indicating a strong upward slope beyond the minimum.
- It crosses the y-axis above the origin.

These graphs illustrate various behaviors of polynomial functions, such as changes in direction, local extrema, and inflection points, which are crucial for understanding calculus concepts like differentiation and integration.

**Analyzing Functions Using Derivatives 5.9**

The derivative of a certain continuous and differentiable function is given by:

\[ y' = 3x^2 - 12x + 9 \]

Analyze the function and create a sketch of the original if the only zeros are at $ x = 0 $ and $ x = 3 $.

*This content is part of the MATH 165, Fall 2020, Online Midterm 1.*

Note: There are no graphs or diagrams provided in the image.

Expert Solution

Step 1: check maxima minima

For maxima-minima, derivative of the function must be zero. That is,

$y' = 0 3 x^{2} - 12 x + 9 = 0 x^{2} - 4 x + 3 = 0 (x - 3) (x - 1) = 0 x = 1, 3$

Now, the function has maxima minima at $x = 1 and x = 3$

Now, second order derivative of the function is given by,

$\begin{array}{rcl} y'' & = & \frac{d}{d x} (3 x^{2} - 12 x + 9) \\ = & 6 x - 12 \end{array}$

Now,

${y''}_{x = 1} = {6 x - 12}_{x = 1} = 6 - 12 = - 6 < 0 Thus, the function has maxima at x = 1 {y''}_{x = 3} = {6 x - 12}_{x = 3} = 18 - 12 = 6 > 0 Thus, the function has minima at x = 3$

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