Unit 5.9 – Connecting a Function, Its First Derivative, and Its Second Derivative
AP® Calculus AB & BC | The Ultimate Synthesis of Derivative Concepts
Why This Matters: This is THE most important synthesis topic in Unit 5! Understanding the deep connections between \(f(x)\), \(f'(x)\), and \(f''(x)\) is essential for mastering calculus. The first derivative tells you about slopes and whether the function is increasing or decreasing. The second derivative tells you about concavity and how the rate of change is changing. Together, they reveal everything about function behavior: extrema, inflection points, shape, and trends. This topic appears frequently on AP® exams, especially in multiple-choice and free-response questions asking you to match graphs, identify features, or justify conclusions using derivative information!
🔗 The Master Connection: \(f\), \(f'\), and \(f''\)
THE COMPLETE RELATIONSHIP CHART
| Information About | What It Tells You | How to Find It | Graph Feature |
|---|---|---|---|
| \(f(x)\) | Function value (height) | Original function | Points on the curve |
| \(f'(x) > 0\) | \(f\) is increasing | Sign of first derivative | Graph rises ↗ |
| \(f'(x) < 0\) | \(f\) is decreasing | Sign of first derivative | Graph falls ↘ |
| \(f'(x) = 0\) | Critical point (horizontal tangent) | Solve \(f'(x) = 0\) | Possible max, min, or horizontal inflection |
| \(f'(x)\) undefined | Critical point (corner/cusp) | Find where \(f'\) DNE | Sharp point or vertical tangent |
| \(f''(x) > 0\) | \(f\) is concave up; \(f'\) is increasing | Sign of second derivative | Curves upward ∪ |
| \(f''(x) < 0\) | \(f\) is concave down; \(f'\) is decreasing | Sign of second derivative | Curves downward ∩ |
| \(f''(x) = 0\) + sign change | Inflection point (concavity changes) | Solve \(f''(x) = 0\), verify sign change | S-curve transition point |
| \(f'(c) = 0, f''(c) > 0\) | Local minimum at \(c\) | Second Derivative Test | Valley ∪ |
| \(f'(c) = 0, f''(c) < 0\) | Local maximum at \(c\) | Second Derivative Test | Peak ∩ |
📊 Combined Behavior Analysis
The Four Fundamental Combinations
| Sign of \(f'\) | Sign of \(f''\) | \(f\) is... | \(f'\) is... | Graph Shape | Description |
|---|---|---|---|---|---|
| + (positive) | + (positive) | Increasing & Concave Up | Increasing | ⤴ | Rising faster and faster |
| + (positive) | − (negative) | Increasing & Concave Down | Decreasing | ⤵ | Rising, but slowing down |
| − (negative) | + (positive) | Decreasing & Concave Up | Increasing | ⤴ | Falling, but slowing down |
| − (negative) | − (negative) | Decreasing & Concave Down | Decreasing | ⤵ | Falling faster and faster |
💡 Key Insight:
- \(f'\) tells you DIRECTION: positive = up, negative = down
- \(f''\) tells you CURVATURE: positive = ∪ (cup), negative = ∩ (cap)
- \(f''\) also tells you about \(f'\): If \(f'' > 0\), then \(f'\) is increasing
- Think of driving: \(f'\) is velocity, \(f''\) is acceleration
🎯 Graph Matching Principles
MATCHING GRAPHS OF \(f\), \(f'\), AND \(f''\)
- Local max/min of \(f\) → Zero (x-intercept) of \(f'\)
- \(f\) increasing → \(f' > 0\) (above x-axis)
- \(f\) decreasing → \(f' < 0\) (below x-axis)
- Inflection point of \(f\) → Local extremum of \(f'\)
- \(f\) concave up → \(f'\) increasing
- \(f\) concave down → \(f'\) decreasing
- Steep slope on \(f\) → Large magnitude of \(f'\)
- Local max/min of \(f'\) → Zero (x-intercept) of \(f''\)
- \(f'\) increasing → \(f'' > 0\) (above x-axis)
- \(f'\) decreasing → \(f'' < 0\) (below x-axis)
- Zero of \(f'\) → Critical point of \(f\)
- Inflection point of \(f\) → Zero (x-intercept) of \(f''\)
- \(f\) concave up → \(f'' > 0\) (above x-axis)
- \(f\) concave down → \(f'' < 0\) (below x-axis)
❓ Critical Analysis Questions
Questions You Must Be Able to Answer:
- If \(f'(2) = 0\) and \(f''(2) > 0\), what can you conclude about \(f\) at \(x = 2\)?
- Answer: \(f\) has a local minimum at \(x = 2\)
- If \(f'(x) > 0\) and \(f''(x) < 0\), what is \(f\) doing?
- Answer: Increasing and concave down (rising but slowing)
- If \(f''(x)\) changes from positive to negative at \(x = c\), what happens at \(x = c\)?
- Answer: \(f\) has an inflection point (concavity changes)
- Can \(f\) be increasing while \(f'\) is decreasing?
- Answer: YES! If \(f' > 0\) but \(f'' < 0\) (increasing concave down)
- If \(f\) has an inflection point at \(x = 3\), what do you know about \(f''\)?
- Answer: \(f''(3) = 0\) or undefined, AND \(f''\) changes sign
📖 Comprehensive Worked Examples
Example 1: Complete Analysis from Function
Problem: Given \(f(x) = x^3 - 3x^2 - 9x + 5\), analyze the complete behavior including connections to \(f'\) and \(f''\).
Solution:
Step 1: Find \(f'(x)\) and analyze
Critical points: \(x = -1, 3\)
| Interval | Sign of \(f'\) | \(f\) Behavior |
|---|---|---|
| \((-\infty, -1)\) | + | Increasing ↗ |
| \((-1, 3)\) | − | Decreasing ↘ |
| \((3, \infty)\) | + | Increasing ↗ |
Extrema: Local max at \(x = -1\), local min at \(x = 3\)
Step 2: Find \(f''(x)\) and analyze
Potential inflection point: \(x = 1\)
| Interval | Sign of \(f''\) | \(f\) Concavity | \(f'\) Behavior |
|---|---|---|---|
| \((-\infty, 1)\) | − | Concave Down ∩ | \(f'\) decreasing |
| \((1, \infty)\) | + | Concave Up ∪ | \(f'\) increasing |
Inflection point: \(x = 1\)
Step 3: Complete Connection Chart
x-values: -1 1 3
f'(x): + + + 0 − − − − − − − 0 + + +
f: Incr MAX Decr Decr MIN Incr
f''(x): − − − − − − − 0 + + + + + + +
f: ∩Down ∩Down INFL ∪Up ∪Up
f': Decr Decr Incr Incr
Combined Analysis:
• (-∞, -1): Increasing + Concave Down → Rising but slowing
• (-1, 1): Decreasing + Concave Down → Falling faster
• (1, 3): Decreasing + Concave Up → Falling but slowing
• (3, ∞): Increasing + Concave Up → Rising faster
Complete Summary:
• Local max at \(x = -1\): \(f'(-1) = 0, f''(-1) < 0\) ✓
• Inflection at \(x = 1\): \(f''(1) = 0\), concavity changes ✓
• Local min at \(x = 3\): \(f'(3) = 0, f''(3) > 0\) ✓
• \(f'\) has max at inflection point of \(f\) ✓
Example 2: Matching Graphs
Problem: Given that \(f(x)\) is a cubic function with a local max at \(x = -2\), local min at \(x = 2\), and inflection point at \(x = 0\), describe the graphs of \(f'(x)\) and \(f''(x)\).
Solution:
Graph of \(f'(x)\):
- Zeros: At \(x = -2\) and \(x = 2\) (where \(f\) has extrema)
- Sign:
- Positive for \(x < -2\) (\(f\) increasing)
- Negative for \(-2 < x < 2\) (\(f\) decreasing)
- Positive for \(x > 2\) (\(f\) increasing)
- Shape: Parabola opening upward (since \(f\) is cubic)
- Extremum: Minimum at \(x = 0\) (where \(f\) has inflection point)
→ \(f'(x)\) looks like an upward parabola with vertex at \((0, \text{negative})\) and zeros at \(x = -2, 2\)
Graph of \(f''(x)\):
- Zero: At \(x = 0\) (where \(f\) has inflection point)
- Sign:
- Negative for \(x < 0\) (\(f\) concave down)
- Positive for \(x > 0\) (\(f\) concave up)
- Shape: Linear (since \(f'\) is quadratic)
→ \(f''(x)\) is a straight line passing through \((0, 0)\) with positive slope
Verification:
If \(f(x) = x^3 - 12x\), then:
• \(f'(x) = 3x^2 - 12 = 3(x^2 - 4) = 3(x-2)(x+2)\) ← Parabola, zeros at ±2 ✓
• \(f''(x) = 6x\) ← Linear through origin ✓
Example 3: Reasoning from Derivative Information
Problem: Suppose \(f'(5) = 0\), \(f''(5) = 0\), \(f'(x) > 0\) for \(x < 5\), and \(f'(x) < 0\) for \(x > 5\). What can you conclude about \(f\) at \(x = 5\)?
Solution:
Analysis:
- \(f'(5) = 0\): Critical point at \(x = 5\)
- \(f'(x) > 0\) for \(x < 5\): \(f\) is increasing before \(x = 5\)
- \(f'(x) < 0\) for \(x > 5\): \(f\) is decreasing after \(x = 5\)
- Sign change of \(f'\): From + to − at \(x = 5\)
Conclusion using First Derivative Test:
Since \(f'\) changes from positive to negative at \(x = 5\), there is a LOCAL MAXIMUM at \(x = 5\).
Note about \(f''(5) = 0\):
The Second Derivative Test is inconclusive when \(f''(c) = 0\). This is why we used the First Derivative Test instead. The fact that \(f''(5) = 0\) might indicate an inflection point near \(x = 5\), but we need more information about sign changes of \(f''\) to confirm.
Answer: \(f\) has a local maximum at \(x = 5\) (by First Derivative Test). The Second Derivative Test cannot be used because \(f''(5) = 0\).
🔍 Special Relationships and Patterns
Key Patterns to Recognize:
At an inflection point of \(f\), the graph of \(f'\) has a local extremum.
- If \(f\) changes from concave down to concave up → \(f'\) has a local minimum
- If \(f\) changes from concave up to concave down → \(f'\) has a local maximum
Each derivative reduces the degree by 1:
- \(f(x)\) is degree \(n\) → \(f'(x)\) is degree \(n-1\) → \(f''(x)\) is degree \(n-2\)
- Cubic \(f\) → Quadratic \(f'\) → Linear \(f''\)
- Quartic \(f\) → Cubic \(f'\) → Quadratic \(f''\)
For a polynomial of degree \(n\):
- At most \(n-1\) critical points (\(f'\) has at most \(n-1\) zeros)
- At most \(n-2\) inflection points (\(f''\) has at most \(n-2\) zeros)
- At most \(n-1\) local extrema
📝 Common AP® Exam Question Types
Typical Questions You'll See:
- Graph Matching: "Which graph represents \(f'\) if the graph of \(f\) is given?"
- Sign Analysis: "For what values of \(x\) is \(f\) increasing and concave down?"
- Critical Point Classification: "Use the given information to determine whether \(f\) has a max or min at \(x = c\)"
- Inflection Points: "How many inflection points does \(f\) have?"
- Justification: "Explain why \(f\) must have a local maximum at \(x = 3\)"
- True/False: "If \(f'(x) > 0\) and \(f''(x) < 0\), is \(f(x)\) increasing? Justify."
- Multiple Representations: "Given a table of values for \(f'\) and \(f''\), describe \(f\)"
Sample AP-Style Question:
Let \(f\) be a function with \(f'(x) = x^2(x-3)\). On what intervals is \(f\) increasing and concave down?
Solution:
Step 1: Analyze \(f'(x)\)
\(f'(x) = x^2(x-3)\) has zeros at \(x = 0, 3\)
- \(x < 0\): \(f'(x) = (+)(-) = - < 0\) → Decreasing
- \(0 < x < 3\): \(f'(x) = (+)(-) = - < 0\) → Decreasing
- \(x > 3\): \(f'(x) = (+)(+) = + > 0\) → Increasing
Step 2: Find and analyze \(f''(x)\)
Zeros at \(x = 0, 2\)
- \(x < 0\): \(f''(x) = (-)(-) = + > 0\) → Concave Up
- \(0 < x < 2\): \(f''(x) = (+)(-) = - < 0\) → Concave Down
- \(x > 2\): \(f''(x) = (+)(+) = + > 0\) → Concave Up
Step 3: Find overlap
Need: \(f' > 0\) (increasing) AND \(f'' < 0\) (concave down)
- Increasing: \((3, \infty)\)
- Concave down: \((0, 2)\)
- Overlap: NONE
Answer: There are no intervals where \(f\) is both increasing and concave down. The function increases on \((3, \infty)\) where it is concave up.
💡 Tips, Tricks & Strategies
✅ Essential Connection Tips:
- Always think hierarchically: \(f'' → f' → f\) (work backwards from second derivative)
- Zeros are key: Zeros of \(f'\) = critical points of \(f\); zeros of \(f''\) = potential inflection points
- Sign determines behavior: Make sign charts for both \(f'\) and \(f''\)
- Check for sign changes: Not every zero is a feature (e.g., \(f''(x) = 0\) without sign change ≠ inflection)
- Use both tests: First Derivative Test always works; Second Derivative Test is faster but can fail
- Match features systematically: Extrema → Zeros, Concavity → Sign, Inflections → Extrema of derivative
- Remember the "levels": Info about \(f''\) tells you about \(f'\) too!
- Draw charts: Visual organization prevents errors
🔥 Quick Recognition Guide:
| If you see... | It means... |
|---|---|
| \(f'\) crosses x-axis | \(f\) has critical point (possible extremum) |
| \(f'\) has extremum | \(f\) has inflection point |
| \(f'\) above x-axis | \(f\) is increasing |
| \(f'\) below x-axis | \(f\) is decreasing |
| \(f'\) increasing | \(f\) is concave up; \(f'' > 0\) |
| \(f'\) decreasing | \(f\) is concave down; \(f'' < 0\) |
| \(f''\) crosses x-axis | \(f\) has inflection point; \(f'\) has extremum |
| \(f''\) above x-axis | \(f\) concave up; \(f'\) increasing |
❌ Common Mistakes to Avoid
- Mistake 1: Confusing zeros of \(f\) with zeros of \(f'\) (x-intercepts vs critical points)
- Mistake 2: Thinking \(f'' > 0\) means \(f\) is increasing (it means concave up!)
- Mistake 3: Assuming every zero of \(f''\) is an inflection point (must verify sign change)
- Mistake 4: Not recognizing that inflection points of \(f\) = extrema of \(f'\)
- Mistake 5: Forgetting that \(f\) can increase while \(f'\) decreases (if \(f' > 0\) but \(f'' < 0\))
- Mistake 6: Matching wrong graphs (e.g., thinking linear \(f\) gives quadratic \(f'\))
- Mistake 7: Not using First Derivative Test when Second Derivative Test is inconclusive
- Mistake 8: Ignoring that \(f''\) tells you about BOTH \(f\) (concavity) and \(f'\) (inc/dec)
- Mistake 9: Making sign errors when testing intervals
- Mistake 10: Not checking ALL conditions (e.g., critical point + sign change for max/min)
📝 Practice Problems
Set A: Sign Analysis
- If \(f'(x) = (x-1)^2(x+2)\), find where \(f\) is increasing and identify extrema.
- If \(f''(x) = x(x-3)\), find inflection points and intervals of concavity for \(f\).
- Given \(f'(2) = 0, f''(2) < 0\), what can you conclude about \(f\) at \(x = 2\)?
Answers:
- Increasing on \((-2, \infty)\); local min at \(x = -2\); no extremum at \(x = 1\)
- Inflection points at \(x = 0, 3\); concave down on \((0,3)\); concave up on \((-\infty,0) \cup (3,\infty)\)
- Local maximum at \(x = 2\) (Second Derivative Test)
Set B: Graph Matching
- If \(f(x) = x^4\), describe the graphs of \(f'(x)\) and \(f''(x)\).
- If \(f'\) is a parabola opening down with zeros at \(x = -1, 3\), describe \(f\).
Answers:
- \(f'(x) = 4x^3\) (cubic through origin); \(f''(x) = 12x^2\) (parabola up, vertex at origin)
- \(f\) has local max at \(x = -1\), local min at \(x = 3\); decreasing on \((-1,3)\)
Set C: Reasoning
- Can \(f'\) be positive while \(f''\) is negative? What does this mean for \(f\)?
- If \(f\) has exactly 2 inflection points, what is the minimum degree of \(f\)?
- True or False: If \(f''(c) = 0\), then \(f\) has an inflection point at \(x = c\). Explain.
Answers:
- Yes! \(f\) is increasing and concave down (rising but slowing down)
- Degree 4 (since \(f''\) needs at least 2 zeros with sign changes)
- False. Need \(f''(c) = 0\) AND sign change of \(f''\). Example: \(f(x) = x^4\) at \(x = 0\)
✏️ AP® Exam Success Tips
What AP® Graders Look For:
- Complete analysis: Address both \(f'\) and \(f''\) when relevant
- Proper justification: State "by First Derivative Test" or "since \(f'' > 0\)"
- Sign analysis shown: Create clear sign charts or test intervals
- Correct terminology: "Local maximum" not just "max"; "inflection point" not "turning point"
- Verification of conditions: Show sign changes, not just zeros
- Clear connections stated: Explicitly connect features (e.g., "zero of \(f'\) indicates...")
- Complete answers: Include intervals, not just isolated points
- Proper reasoning: Explain WHY, not just WHAT
💯 Scoring Maximum Points:
- Show work: Sign charts, test points, calculations visible
- State theorems: Name the test you're using
- Verify all conditions: Don't assume—check everything
- Use proper notation: Interval notation, inequality signs correctly
- Answer completely: If asked for intervals AND points, give both
- Justify conclusions: Every claim needs evidence from derivatives
⚡ Ultimate Quick Reference Card
| Derivative Info | Conclusion about \(f\) | Conclusion about \(f'\) |
|---|---|---|
| \(f' > 0\) | \(f\) increasing | — |
| \(f' < 0\) | \(f\) decreasing | — |
| \(f' = 0\) | Critical point | — |
| \(f'' > 0\) | \(f\) concave up | \(f'\) increasing |
| \(f'' < 0\) | \(f\) concave down | \(f'\) decreasing |
| \(f'' = 0\) + sign change | Inflection point | \(f'\) has extremum |
| \(f' = 0, f'' > 0\) | Local minimum | — |
| \(f' = 0, f'' < 0\) | Local maximum | — |
| \(f' > 0, f'' > 0\) | Increasing, concave up | \(f'\) increasing |
| \(f' > 0, f'' < 0\) | Increasing, concave down | \(f'\) decreasing |
Master the Connections! This synthesis topic brings together ALL of Unit 5: the first derivative \(f'(x)\) reveals slopes, critical points (where \(f' = 0\) or undefined), and whether \(f\) is increasing (\(f' > 0\)) or decreasing (\(f' < 0\)); the second derivative \(f''(x)\) reveals concavity (positive = ∪ up, negative = ∩ down), inflection points (where \(f''\) changes sign), and whether \(f'\) is increasing or decreasing. Together they classify extrema: First Derivative Test uses sign changes of \(f'\), Second Derivative Test uses sign of \(f''\) at critical points. Remember key connections: zeros of \(f'\) are critical points of \(f\); extrema of \(f'\) are inflection points of \(f\); sign of \(f'\) determines inc/dec of \(f\); sign of \(f''\) determines concavity of \(f\) AND inc/dec of \(f'\). Create sign charts for both derivatives, verify sign changes, use proper justification, and always connect features systematically. This is THE most tested synthesis topic on AP® exams! 🎯✨