Lesson: Interpreting Graphs of Derivatives | Nagwa - Free Printable

The image provided is about interpreting graphs of derivatives. The task involves analyzing the graphs of \( y = f(x) \), \( y = f'(x) \), and \( y = f''(x) \) to determine critical points, points of inflection, concavity, and whether the function is increasing or decreasing. Let's break this down step by step:

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1. Critical Points

- Definition: Critical points occur where the first derivative \( f'(x) \) is zero or undefined.
- Graph Analysis:
- From the graph of \( y = f'(x) \) (the orange curve):
- The curve crosses the x-axis at one point. This indicates that \( f'(x) = 0 \) at that point.
- There are no vertical asymptotes or other undefined points in the graph of \( f'(x) \).
- Conclusion: The critical point occurs where the orange curve intersects the x-axis.

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2. Points of Inflection

- Definition: Points of inflection occur where the second derivative \( f''(x) \) changes sign, indicating a change in concavity.
- Graph Analysis:
- From the graph of \( y = f''(x) \) (the pink line):
- The pink line is a straight line with a positive slope, which means \( f''(x) \) is always positive and never changes sign.
- Conclusion: There are no points of inflection because \( f''(x) \) does not change sign.

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3. Concavity

- Definition: Concavity is determined by the sign of the second derivative \( f''(x) \):
- If \( f''(x) > 0 \), the function is concave up.
- If \( f''(x) < 0 \), the function is concave down.
- Graph Analysis:
- From the graph of \( y = f''(x) \) (the pink line):
- The pink line is above the x-axis for all \( x \), indicating that \( f''(x) > 0 \) everywhere.
- Conclusion: The function \( f(x) \) is concave up everywhere.

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4. Increasing or Decreasing

- Definition: The behavior of the function (increasing or decreasing) is determined by the sign of the first derivative \( f'(x) \):
- If \( f'(x) > 0 \), the function is increasing.
- If \( f'(x) < 0 \), the function is decreasing.
- Graph Analysis:
- From the graph of \( y = f'(x) \) (the orange curve):
- The orange curve is below the x-axis for \( x < 0 \) and above the x-axis for \( x > 0 \).
- At \( x = 0 \), the orange curve crosses the x-axis, indicating \( f'(0) = 0 \).
- Conclusion:
- The function \( f(x) \) is decreasing for \( x < 0 \) (where \( f'(x) < 0 \)).
- The function \( f(x) \) is increasing for \( x > 0 \) (where \( f'(x) > 0 \)).

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Summary of Observations

1. Critical Points: Occur where \( f'(x) = 0 \). From the graph of \( f'(x) \), there is one critical point.
2. Points of Inflection: Occur where \( f''(x) \) changes sign. From the graph of \( f''(x) \), there are no points of inflection.
3. Concavity: The function is concave up everywhere because \( f''(x) > 0 \) for all \( x \).
4. Increasing or Decreasing: The function is decreasing for \( x < 0 \) and increasing for \( x > 0 \).

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Final Answer

\[
\boxed{
\begin{aligned}
&\text{1. Critical points: One critical point where } f'(x) = 0. \\
&\text{2. Points of inflection: No points of inflection.} \\
&\text{3. Concavity: The function is concave up everywhere.} \\
&\text{4. Increasing or Decreasing: Decreasing for } x < 0 \text{ and increasing for } x > 0.
\end{aligned}
}
\]