Sketch a function that is continuous on left parenthesis negative infinity comma infinity right parenthesis and has the following properties. Use a number line to summarize information about the function.
f prime left parenthesis x right parenthesisless than0 on left parenthesis negative infinity comma negative 2 right parenthesis​; f prime left parenthesis x right parenthesisgreater than0 on left parenthesis negative 2 comma 3 right parenthesis​; f prime left parenthesis x right parenthesisless than0 on left parenthesis 3 comma infinity right parenthesis.
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Part 1
Which of the following graphs matches the description of the given​ properties?
A.
-10
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-20
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y



A coordinate system has a horizontal x-axis labeled from negative 10 to 10 in increments of 5 and vertical y-axis from negative 20 to 20 in increments of 5. From left to right, a smooth curve falls, crossing the x-axis at negative 2, to a minimum, and rises, crossing the x-axis at 3.
B.
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y



A coordinate system has a horizontal x-axis labeled from negative 10 to 10 in increments of 5 and vertical y-axis from negative 20 to 20 in increments of 5. From left to right, a smooth curve rises, crossing the x-axis at negative 2, to a maximum, and falls, crossing the x-axis at 3.
C.
-10
10
-20
20
x
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A coordinate system has a horizontal x-axis labeled from negative 10 to 10 in increments of 5 and vertical y-axis from negative 20 to 20 in increments of 5. From left to right, a smooth curve falls to a minimum at x = negative 2, rises to a maximum at x = 3, and falls.
D.
-10
10
-20
20
x
y




To determine which graph matches the given properties, let’s analyze the behavior of \( f'(x) \) in each interval and how that affects the shape of \( f(x) \).

### Given Conditions:
1. \( f'(x) < 0 \) on \( (-\infty, -2) \): \( f(x) \) is **decreasing** in this interval.
2. \( f'(x) > 0 \) on \( (-2, 3) \): \( f(x) \) is **increasing** in this interval.
3. \( f'(x) < 0 \) on \( (3, \infty) \): \( f(x) \) is **decreasing** in this interval.

### Implications:
- At \( x = -2 \): The function switches from decreasing to increasing, indicating a **local minimum**.
- At \( x = 3 \): The function switches from increasing to decreasing, indicating a **local maximum**.

### Analyzing Each Option:
- **Option A**: The function decreases, reaches a minimum, and then increases through \( x = -2 \), and continues to increase through \( x = 3 \). This does not match the conditions because it lacks a maximum at \( x = 3 \).
- **Option B**: The function increases, reaches a maximum at \( x = -2 \), and then decreases through \( x = 3 \). This does not match the conditions because it has a maximum at \( x = -2 \) rather than a minimum.
- **Option C**: The function decreases to a minimum at \( x = -2 \), increases to a maximum at \( x = 3 \), and then decreases again. This matches all given conditions.
- **Option D**: The graph does not match the conditions as it doesn’t clearly show the correct changes in direction.

### Conclusion:
The correct answer is **Option C**. It aligns with the requirement that \( f(x) \) has a local minimum at \( x = -2 \), a local maximum at \( x = 3 \), and the correct intervals of increase and decrease.

Would you like a more detailed explanation of why Option C is correct? Here are five related questions to expand on this concept:

1. How does the sign of \( f'(x) \) affect the shape of \( f(x) \) in general?
2. What would happen if \( f'(x) \) was positive on \( (-\infty, -2) \)?
3. How would the graph change if \( f'(x) \) was zero at \( x = -2 \) and \( x = 3 \)?
4. Can a function have both a maximum and minimum at the same point?
5. How would this analysis change if we were given information about \( f''(x) \)?

**Tip:** To confirm the nature of extrema, use the first or second derivative test at key points.

Sketch a function that is continuous on (-∞, ∞) and has the following properties. Use a number line to summarize information about the function.
- f'(x) < 0 on (-∞, -2)
- f'(x) > 0 on (-2, 3)
- f'(x) < 0 on (3, ∞)

Which of the following graphs matches the description of the given properties?

Explore how to analyze a continuous function with specified increasing and decreasing intervals. Using the first derivative test, this example helps to identify local minima and maxima points and match a suitable graph, perfect for high school calculus students.

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