Unit 5.6 – Determining Concavity of Functions over Their Domains

AP® Calculus AB & BC | Understanding Concavity and the Second Derivative

Why This Matters: Concavity describes how a function curves—whether it bends upward like a smile (∪) or downward like a frown (∩). Understanding concavity is essential for analyzing function behavior, sketching accurate graphs, and identifying inflection points where the curve changes direction. The second derivative \(f''(x)\) is the key tool: it tells us about the rate of change of the rate of change! This concept is crucial for optimization, understanding acceleration in physics, and analyzing economic models. Mastering concavity completes your understanding of how derivatives reveal function behavior!

📐 Definitions: Concave Up and Concave Down

FORMAL DEFINITIONS

Concave Up (Concave Upward)

A function \(f\) is concave up on an interval \(I\) if the graph of \(f\) lies above all of its tangent lines on that interval.

Visual: The graph curves upward like a cup or smile: ∪

Slope behavior: The slopes of tangent lines are increasing from left to right.

Concave Down (Concave Downward)

A function \(f\) is concave down on an interval \(I\) if the graph of \(f\) lies below all of its tangent lines on that interval.

Visual: The graph curves downward like a cap or frown: ∩

Slope behavior: The slopes of tangent lines are decreasing from left to right.

💡 Memory Tricks:

Concave UP: Opens upward like a CUP ∪ (holds water)
Concave DOWN: Opens downward like a frowN ∩ (spills water)
Concave UP: Smiling face 😊 (curves up at ends)
Concave DOWN: Sad face ☹️ (curves down at ends)

🎯 The Concavity Test (Second Derivative Test)

Test for Concavity

Let \(f\) be a function whose second derivative exists on an interval \(I\):

If \(f''(x) > 0\) for all \(x\) in \(I\)

\[ \text{Then } f \text{ is CONCAVE UP on } I \]

Positive second derivative → Concave up (∪)

If \(f''(x) < 0\) for all \(x\) in \(I\)

\[ \text{Then } f \text{ is CONCAVE DOWN on } I \]

Negative second derivative → Concave down (∩)

🔑 The Key Connection:

Second Derivative and Concavity
Sign of \(f''(x)\)	Concavity	Shape	\(f'(x)\) Behavior
\(f''(x) > 0\)	Concave Up	∪ (cup/smile)	\(f'\) is increasing
\(f''(x) < 0\)	Concave Down	∩ (cap/frown)	\(f'\) is decreasing
\(f''(x) = 0\)	Possible inflection point	Change in concavity	\(f'\) has critical point

📝 Understanding the Connection:

\(f'(x)\) tells you the slope of the tangent line
\(f''(x)\) tells you how the slope is changing
If \(f'' > 0\), slopes are increasing → graph curves upward (concave up)
If \(f'' < 0\), slopes are decreasing → graph curves downward (concave down)

🔄 Inflection Points

DEFINITION: INFLECTION POINT

A point \((c, f(c))\) is an inflection point of \(f\) if:

\(f\) is continuous at \(x = c\)
The concavity changes at \(x = c\)
- From concave up to concave down, OR
- From concave down to concave up

In words: An inflection point is where the curve changes from bending one way to bending the other way.

Finding Potential Inflection Points

Possible inflection points occur where:

\[ f''(x) = 0 \quad \text{OR} \quad f''(x) \text{ is undefined} \]

BUT: You must verify that concavity actually changes!

⚠️ Critical Warning:

NOT every point where \(f''(x) = 0\) is an inflection point!

Example: \(f(x) = x^4\) has \(f''(0) = 0\), but \(x = 0\) is NOT an inflection point
The concavity does not change—it's concave up on both sides!
Always verify that \(f''\) changes sign at the point

📊 Inflection Point Test:

To verify \(x = c\) is an inflection point:

Check that \(f\) is continuous at \(x = c\)
Find where \(f''(x) = 0\) or \(f''(x)\) is undefined
Test the sign of \(f''(x)\) on intervals around \(c\)
If \(f''\) changes sign, then \(x = c\) is an inflection point
If \(f''\) does NOT change sign, then \(x = c\) is NOT an inflection point

📋 Step-by-Step Procedure: Finding Intervals of Concavity

The Complete Process:

Find the first derivative \(f'(x)\)
Find the second derivative \(f''(x)\)
Find potential inflection points:
- Solve \(f''(x) = 0\)
- Find where \(f''(x)\) is undefined (but \(f(x)\) is defined)
Create a number line marking all potential inflection points
Choose test points in each interval
Evaluate the sign of \(f''(\text{test point})\)
Determine concavity:
- If \(f''(x) > 0\) → Concave up
- If \(f''(x) < 0\) → Concave down
Identify inflection points where \(f''\) changes sign
Write intervals using proper notation

📖 Comprehensive Worked Examples

Example 1: Finding Concavity and Inflection Points

Problem: Find the intervals of concavity and inflection points of \(f(x) = x^3 - 6x^2 + 9x + 1\).

Solution:

Step 1: Find \(f'(x)\)

\[ f'(x) = 3x^2 - 12x + 9 \]

Step 2: Find \(f''(x)\)

\[ f''(x) = 6x - 12 = 6(x - 2) \]

Step 3: Find potential inflection points

Set \(f''(x) = 0\):

\[ 6(x - 2) = 0 \quad \Rightarrow \quad x = 2 \]

\(f''(x)\) is never undefined (polynomial)

Potential inflection point: \(x = 2\)

Step 4: Test intervals around \(x = 2\)

Interval	Test Point	\(f''(\text{test})\)	Sign	Concavity
\((-\infty, 2)\)	\(x = 0\)	\(6(0) - 12 = -12\)	−	Concave Down
\((2, \infty)\)	\(x = 3\)	\(6(3) - 12 = 6\)	+	Concave Up

Step 5: Create sign chart

Potential inflection:       2

f''(x) sign:     − − − −  0  + + + +

Concavity:       ∩ Down     ∪ Up

Sign changes:        −→+
                 INFLECTION POINT

Step 6: Verify inflection point and find coordinates

Since \(f''\) changes sign from − to + at \(x = 2\), it IS an inflection point ✓

Calculate \(f(2)\):

\[ f(2) = 2^3 - 6(2)^2 + 9(2) + 1 = 8 - 24 + 18 + 1 = 3 \]

Final Answer:
• Concave down: \((-\infty, 2)\)
• Concave up: \((2, \infty)\)
• Inflection point: \((2, 3)\)

Example 2: Function with Multiple Inflection Points

Problem: Find intervals of concavity and inflection points of \(f(x) = x^4 - 4x^3\).

Solution:

Step 1-2: Find \(f'(x)\) and \(f''(x)\)

\[ f'(x) = 4x^3 - 12x^2 \]

\[ f''(x) = 12x^2 - 24x = 12x(x - 2) \]

Step 3: Find potential inflection points

\[ 12x(x - 2) = 0 \]

\[ x = 0 \quad \text{or} \quad x = 2 \]

Step 4: Test intervals

Interval	Test Point	\(f''(x) = 12x(x-2)\)	Sign	Concavity
\((-\infty, 0)\)	\(x = -1\)	\(12(-1)(-3) = 36\)	+	Concave Up
\((0, 2)\)	\(x = 1\)	\(12(1)(-1) = -12\)	−	Concave Down
\((2, \infty)\)	\(x = 3\)	\(12(3)(1) = 36\)	+	Concave Up

Step 5: Sign chart

Potential points:        0          2

f''(x) sign:    + + + +  0  − − − −  0  + + + +

Concavity:      ∪ Up     ∩ Down     ∪ Up

Sign changes:       +→−          −→+
                   INFL.        INFL.

Step 6: Verify and find coordinates

At \(x = 0\): \(f''\) changes from + to − → Inflection point ✓
- \(f(0) = 0\)
At \(x = 2\): \(f''\) changes from − to + → Inflection point ✓
- \(f(2) = 16 - 32 = -16\)

Final Answer:
• Concave up: \((-\infty, 0) \cup (2, \infty)\)
• Concave down: \((0, 2)\)
• Inflection points: \((0, 0)\) and \((2, -16)\)

Example 3: Point Where \(f''(x) = 0\) But NO Inflection Point

Problem: Analyze \(f(x) = x^4\) for concavity and inflection points.

Solution:

Step 1-2: Find derivatives

\[ f'(x) = 4x^3 \]

\[ f''(x) = 12x^2 \]

Step 3: Find potential inflection points

\[ 12x^2 = 0 \quad \Rightarrow \quad x = 0 \]

Step 4: Test intervals around \(x = 0\)

Interval	Test Point	\(f''(x) = 12x^2\)	Sign
\((-\infty, 0)\)	\(x = -1\)	\(12(1) = 12\)	+
\((0, \infty)\)	\(x = 1\)	\(12(1) = 12\)	+

Step 5: Analyze sign change

                        0

f''(x) sign:    + + + +  0  + + + +

Concavity:      ∪ Up       ∪ Up

Sign change:       NONE!

\(f''\) does NOT change sign → NO inflection point at \(x = 0\)

Final Answer:
• Concave up: \((-\infty, \infty)\) (entire domain)
• Concave down: Never
• Inflection points: None
(Even though \(f''(0) = 0\), it's not an inflection point!)

Example 4: Function with Undefined Second Derivative

Problem: Find concavity and inflection points of \(f(x) = x^{1/3}\).

Solution:

Step 1-2: Find derivatives

\[ f'(x) = \frac{1}{3}x^{-2/3} = \frac{1}{3x^{2/3}} \]

\[ f''(x) = -\frac{2}{9}x^{-5/3} = -\frac{2}{9x^{5/3}} \]

Step 3: Find potential inflection points

\(f''(x) = 0\): Never equals zero (numerator is constant −2)
\(f''(x)\) undefined: When \(x^{5/3} = 0\) → \(x = 0\)
Check: Is \(f(0)\) defined? Yes: \(f(0) = 0\) ✓

Potential inflection point: \(x = 0\)

Step 4: Test intervals

Interval	Test Point	\(f''(x) = -\frac{2}{9x^{5/3}}\)	Sign	Concavity
\((-\infty, 0)\)	\(x = -1\)	\(-\frac{2}{9(-1)} = \frac{2}{9}\)	+	Concave Up
\((0, \infty)\)	\(x = 1\)	\(-\frac{2}{9(1)} = -\frac{2}{9}\)	−	Concave Down

Step 5: Analyze

\(f''\) changes from + to − at \(x = 0\) → Inflection point ✓

Final Answer:
• Concave up: \((-\infty, 0)\)
• Concave down: \((0, \infty)\)
• Inflection point: \((0, 0)\)
(Note: This inflection point has a vertical tangent line!)

Example 5: Trigonometric Function

Problem: Find concavity and inflection points of \(f(x) = \sin(x)\) on \([0, 2\pi]\).

Solution:

Step 1-2: Find derivatives

\[ f'(x) = \cos(x) \]

\[ f''(x) = -\sin(x) \]

Step 3: Find potential inflection points

\[ -\sin(x) = 0 \quad \Rightarrow \quad \sin(x) = 0 \]

In \([0, 2\pi]\): \(x = 0, \pi, 2\pi\)

Step 4: Test intervals

Interval	Test Point	\(f''(x) = -\sin(x)\)	Sign	Concavity
\((0, \pi)\)	\(x = \frac{\pi}{2}\)	\(-\sin(\frac{\pi}{2}) = -1\)	−	Concave Down
\((\pi, 2\pi)\)	\(x = \frac{3\pi}{2}\)	\(-\sin(\frac{3\pi}{2}) = 1\)	+	Concave Up

Final Answer:
• Concave down: \((0, \pi)\)
• Concave up: \((\pi, 2\pi)\)
• Inflection point: \((\pi, 0)\)
(Note: \(x = 0\) and \(x = 2\pi\) are endpoints, not inflection points in this interval)

📊 The Complete Picture: \(f\), \(f'\), and \(f''\)

Understanding the Relationships:

Complete Function Behavior Analysis
\(f'(x)\)	\(f''(x)\)	\(f(x)\) Behavior	Graph Shape
+ (positive)	+ (positive)	Increasing & Concave Up	Rising curve ⤴
+ (positive)	− (negative)	Increasing & Concave Down	Rising but leveling off ⤵
− (negative)	+ (positive)	Decreasing & Concave Up	Falling but leveling off ⤴
− (negative)	− (negative)	Decreasing & Concave Down	Falling curve ⤵
0	+ (positive)	Horizontal tangent, Concave Up	Local minimum
0	− (negative)	Horizontal tangent, Concave Down	Local maximum

💡 Quick Reference:

\(f(x)\): The function value (height)
\(f'(x)\): Rate of change (slope) → tells if increasing/decreasing
\(f''(x)\): Rate of change of slope → tells concavity
\(f' > 0\): \(f\) is increasing
\(f' < 0\): \(f\) is decreasing
\(f'' > 0\): \(f\) is concave up (and \(f'\) is increasing)
\(f'' < 0\): \(f\) is concave down (and \(f'\) is decreasing)

🎯 Second Derivative Test for Local Extrema

Second Derivative Test

Suppose \(f'(c) = 0\) (critical point). Then:

If \(f''(c) > 0\)

\[ f(c) \text{ is a LOCAL MINIMUM} \]

Reasoning: Function is concave up at \(c\), so horizontal tangent is at a valley bottom.

If \(f''(c) < 0\)

\[ f(c) \text{ is a LOCAL MAXIMUM} \]

Reasoning: Function is concave down at \(c\), so horizontal tangent is at a hilltop.

If \(f''(c) = 0\) or undefined

\[ \text{TEST IS INCONCLUSIVE} \]

Use First Derivative Test instead!

First Derivative Test vs Second Derivative Test:

Aspect	First Derivative Test	Second Derivative Test
What to find	Sign of \(f'\) around critical point	Sign of \(f''\) at critical point
Works when	Always (if \(f\) continuous)	Only when \(f''(c) \neq 0\)
Advantage	Always reliable	Faster (single evaluation)
Disadvantage	Need to test intervals	Fails when \(f''(c) = 0\)
When to use	Default choice; always works	Quick check if \(f''\) is easy to find

💡 Tips, Tricks & Strategies

✅ Essential Problem-Solving Tips:

Always find \(f''\) correctly: Take the derivative of \(f'\), not \(f\)
Factor \(f''(x)\) completely: Makes finding zeros much easier
Test between potential inflection points: Not at them!
Verify sign changes: \(f''(x) = 0\) alone doesn't guarantee inflection point
Draw sign charts: Visual organization prevents errors
Use proper interval notation: Open intervals for concavity
Include coordinates for inflection points: \((c, f(c))\), not just \(x = c\)
Check continuity: Inflection points require \(f\) to be continuous

🎯 The Ultimate Concavity Checklist:

☑ Find \(f'(x)\)
☑ Find \(f''(x)\) and simplify
☑ Factor \(f''(x)\) completely
☑ Solve \(f''(x) = 0\)
☑ Find where \(f''(x)\) is undefined
☑ Mark all potential inflection points on number line
☑ Test sign of \(f''\) in each interval
☑ Create organized sign chart
☑ Determine concavity in each interval
☑ Verify which points are actual inflection points
☑ Calculate \(f(c)\) for each inflection point
☑ State intervals and points clearly

🔥 Quick Recognition Patterns:

Linear functions: No concavity (\(f'' = 0\) everywhere)
Quadratic \(ax^2 + bx + c\):
- If \(a > 0\): concave up everywhere (∪)
- If \(a < 0\): concave down everywhere (∩)
Cubic functions: Usually one inflection point
Quartic \(x^4\): Concave up everywhere
\(\sin(x)\) and \(\cos(x)\): Alternate concavity every \(\pi\) units
Exponential \(e^x\): Concave up everywhere
Logarithm \(\ln(x)\): Concave down everywhere (for \(x > 0\))

❌ Common Mistakes to Avoid

Mistake 1: Assuming every point where \(f''(x) = 0\) is an inflection point (must verify sign change!)
Mistake 2: Confusing \(f'(x)\) with \(f''(x)\) when determining concavity
Mistake 3: Testing AT potential inflection points instead of BETWEEN them
Mistake 4: Forgetting to check where \(f''(x)\) is undefined
Mistake 5: Not factoring \(f''(x)\) before solving (makes it harder!)
Mistake 6: Stating inflection points as x-values only (need \((x, y)\) coordinates)
Mistake 7: Confusing concave up/down with increasing/decreasing
Mistake 8: Using Second Derivative Test when \(f''(c) = 0\) (inconclusive!)
Mistake 9: Including potential inflection points in concavity intervals
Mistake 10: Not simplifying \(f''(x)\) before analysis
Mistake 11: Mixing up the memory tricks (cup/cap)
Mistake 12: Forgetting that concavity requires \(f\) to be continuous

📝 Practice Problems

Set A: Finding Concavity

Find concavity intervals for \(f(x) = x^3 - 3x + 1\)
Find concavity intervals for \(f(x) = x^4 - 4x^2\)
Find concavity intervals for \(f(x) = e^{-x^2}\)

Answers:

Concave down: \((-\infty, 0)\); Concave up: \((0, \infty)\); Inflection: \((0, 1)\)
Concave up: \((-\infty, -\frac{\sqrt{2}}{2}) \cup (\frac{\sqrt{2}}{2}, \infty)\); Down: \((-\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2})\)
Concave down: \((-1, 1)\); Concave up: \((-\infty, -1) \cup (1, \infty)\)

Set B: Inflection Points

Find all inflection points of \(f(x) = x^4 - 6x^2 + 8\)
Show that \(f(x) = x^6\) has no inflection points
Find inflection points of \(f(x) = x^{5/3}\)

Answers:

Inflection points at \(x = \pm 1\): \((1, 3)\) and \((-1, 3)\)
\(f''(x) = 30x^4 \geq 0\) always (no sign change); no inflection points
Inflection point at \((0, 0)\) (\(f''\) undefined and changes sign)

Set C: Second Derivative Test

Use Second Derivative Test to classify critical points of \(f(x) = x^3 - 3x^2 + 4\)
Explain why Second Derivative Test fails for \(f(x) = x^3\) at \(x = 0\)

Answers:

Critical points: \(x = 0, 2\); \(f''(0) = -6 < 0\) → local max; \(f''(2) = 6 > 0\) → local min
\(f'(0) = 0\) but \(f''(0) = 0\) → test inconclusive. Use First Derivative Test instead.

✏️ AP® Exam Success Tips

What AP® Graders Look For:

Both derivatives shown: Write \(f'(x) = \ldots\) and \(f''(x) = \ldots\)
Potential inflection points identified: Show solving \(f''(x) = 0\)
Sign analysis of \(f''\): Test intervals and show work
Sign chart or number line: Visual organization
Verification of sign changes: Explicitly state where \(f''\) changes sign
Proper interval notation: Use open intervals for concavity
Complete coordinates: Inflection points as \((x, y)\) pairs
Clear conclusions: "Concave up on..." and "Inflection point at..."

Common FRQ Formats:

"Find the intervals on which f is concave up"
"Find all inflection points of f. Justify your answer."
"Use the second derivative to determine..."
"At what x-values does the graph of f change concavity?"
"Given f'(x), determine where f is concave down"
"Justify using the sign of f''"

💯 Earning Full Credit:

1 point: Finding \(f''(x)\) correctly
1 point: Finding potential inflection points
2 points: Sign analysis (1 for attempting, 1 for correct)
1 point: Correct intervals with proper notation
1 point: Identifying and justifying inflection points

⚡ Quick Reference Card

Concavity Quick Reference
Concept	Key Information
Concave Up	\(f''(x) > 0\) → Graph curves upward ∪ → \(f'\) increasing
Concave Down	\(f''(x) < 0\) → Graph curves downward ∩ → \(f'\) decreasing
Inflection Point	Where concavity changes (\(f''\) changes sign)
Finding Inflection Points	Solve \(f''(x) = 0\) or \(f''\) undefined, then verify sign change
Second Derivative Test	\(f'(c) = 0\) and \(f''(c) > 0\) → local min \(f'(c) = 0\) and \(f''(c) < 0\) → local max
Key Warning	\(f''(x) = 0\) doesn't guarantee inflection point!
Memory Aid	Concave UP = CUP ∪ \| Concave DOWN = frowN ∩

🔗 Connections to Other Topics

Topic 5.6 Connects To:

Topic 5.3 (Inc/Dec): \(f' > 0\) means increasing; \(f'' > 0\) means \(f'\) is increasing
Topic 5.4 (1st Deriv Test): Identifies local extrema; 2nd deriv test is alternative
Topic 5.5 (Optimization): Concavity helps verify max/min
Topic 5.7 (Curve Sketching): Concavity determines graph shape
Related Rates: \(f''\) represents acceleration when \(f'\) is velocity
Physics: Acceleration, force, jerk (3rd derivative)
Economics: Marginal cost increasing/decreasing

Master Concavity! The second derivative \(f''(x)\) reveals how a function curves: \(f'' > 0\) means concave up (∪ like a cup), and \(f'' < 0\) means concave down (∩ like a cap). Inflection points occur where concavity changes—find them by solving \(f''(x) = 0\) or finding where \(f''\) is undefined, then verify the sign change. Remember: not every point where \(f''(x) = 0\) is an inflection point! The Second Derivative Test provides a quick way to classify critical points: if \(f'(c) = 0\) and \(f''(c) > 0\), it's a local min; if \(f''(c) < 0\), it's a local max. Use sign charts, test between potential inflection points, and always verify sign changes. Understanding concavity completes your analysis of function behavior and is essential for curve sketching and optimization! 🎯✨