Suppose that f(x) has one critical point, at x = -5. Suppose f'(x) also has the following -6 -5 -4 f'(x) + 0 + x Which of the following could be a graph of y = f(x)? 10 10 10+ -10+ 10+ -10+ IC 10 10 10 10 -10+ 10+ -10+ IC IC
Suppose that f(x) has one critical point, at x = -5. Suppose f'(x) also has the following -6 -5 -4 f'(x) + 0 + x Which of the following could be a graph of y = f(x)? 10 10 10+ -10+ 10+ -10+ IC 10 10 10 10 -10+ 10+ -10+ IC IC
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
![**Problem Statement:**
Suppose that \( f(x) \) has one critical point, at \( x = -5 \). Suppose \( f'(x) \) also has the following values:
\[
\begin{array}{c|c|c|c}
x & -6 & -5 & -4 \\
\hline
f'(x) & + & 0 & + \\
\end{array}
\]
Which of the following could be a graph of \( y = f(x) \)?
**Explanation of Graphs:**
There are four graphs presented, each with different behavior around the point \( x = -5 \), which corresponds to the critical point:
1. **Top Left Graph:**
- Starts from positive \( y \)-values and decreases through the critical point.
- Has a minimum point around \( x = -5 \), then starts increasing.
2. **Top Right Graph:**
- Starts with increasing \( y \)-values, passes through a minimum, and continues increasing.
- The transition is smooth around the critical point.
3. **Bottom Left Graph:**
- Shows an inverted U-shape.
- Indicates no minimum or maximum at the critical point, unlikely to represent the given derivative information.
4. **Bottom Right Graph:**
- Displays a U-shape, indicating a minimum around \( x = -5 \).
- Indicates increasing slope on either side of the critical point.
The key is to identify the graph that matches the derivative behavior:
- At \( x = -5 \), \( f'(x) = 0 \) indicates a horizontal tangent, suggesting a local minimum or maximum.
- Positive derivative signs on either side (\( x = -6 \) and \( x = -4 \)) suggest a local minimum at \( x = -5 \).
Based on the above analysis, the **bottom right graph** is likely the correct graph of \( y = f(x) \), as it demonstrates a local minimum at \( x = -5 \) with a horizontal tangent and increasing values on either side.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F875d9c65-29a5-41bb-98c8-51e0caff1641%2F46754c8c-bdef-4713-8b39-96e79e503cec%2Fbn450w_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Suppose that \( f(x) \) has one critical point, at \( x = -5 \). Suppose \( f'(x) \) also has the following values:
\[
\begin{array}{c|c|c|c}
x & -6 & -5 & -4 \\
\hline
f'(x) & + & 0 & + \\
\end{array}
\]
Which of the following could be a graph of \( y = f(x) \)?
**Explanation of Graphs:**
There are four graphs presented, each with different behavior around the point \( x = -5 \), which corresponds to the critical point:
1. **Top Left Graph:**
- Starts from positive \( y \)-values and decreases through the critical point.
- Has a minimum point around \( x = -5 \), then starts increasing.
2. **Top Right Graph:**
- Starts with increasing \( y \)-values, passes through a minimum, and continues increasing.
- The transition is smooth around the critical point.
3. **Bottom Left Graph:**
- Shows an inverted U-shape.
- Indicates no minimum or maximum at the critical point, unlikely to represent the given derivative information.
4. **Bottom Right Graph:**
- Displays a U-shape, indicating a minimum around \( x = -5 \).
- Indicates increasing slope on either side of the critical point.
The key is to identify the graph that matches the derivative behavior:
- At \( x = -5 \), \( f'(x) = 0 \) indicates a horizontal tangent, suggesting a local minimum or maximum.
- Positive derivative signs on either side (\( x = -6 \) and \( x = -4 \)) suggest a local minimum at \( x = -5 \).
Based on the above analysis, the **bottom right graph** is likely the correct graph of \( y = f(x) \), as it demonstrates a local minimum at \( x = -5 \) with a horizontal tangent and increasing values on either side.
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