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Second Derivative Test
Determining Extrema

Learn how the concavity of a function at a critical point reveals whether you have a local maximum, local minimum, or neither.

AP Calculus BC

What is a Critical Point?

A critical point of a function \( f \) occurs at \( x = c \) when either \( f'(c) = 0 \) or \( f'(c) \) does not exist. These are the only candidates for local extrema.

\( f'(c) = 0 \)
Horizontal tangent: the function levels out at \( x = c \).
\( f'(c) \) DNE
Cusp or corner: the derivative is undefined but an extremum may still occur.

Understanding Concavity

The second derivative \( f''(x) \) measures the rate of change of the slope. It tells us whether the curve bends upward (concave up) or downward (concave down).

Concave Up \( f''(x) > 0 \) min Concave Down \( f''(x) < 0 \) max "holds water" "spills water"

Why Use the Second Derivative?

The First Derivative Test requires checking sign changes of \( f' \) on intervals around the critical point. The Second Derivative Test gives an answer with a single evaluation of \( f'' \) at the critical point.

Efficiency
One computation instead of testing intervals on both sides.
Clarity
Directly links the geometry of the curve to the conclusion.
Limitation
Inconclusive when \( f''(c) = 0 \): must fall back to another method.

The Second Derivative Test

Let \( c \) be a critical point of \( f \) where \( f'(c) = 0 \) and \( f'' \) is continuous near \( c \):

1
If \( f''(c) > 0 \), then \( f \) has a local minimum at \( x = c \).
2
If \( f''(c) < 0 \), then \( f \) has a local maximum at \( x = c \).
3
If \( f''(c) = 0 \), the test is inconclusive: use the First Derivative Test or higher-order derivatives.

Visualizing the Three Cases

Local Min \( f''(c) > 0 \) tangent flat Local Max \( f''(c) < 0 \) tangent flat Inconclusive \( f''(c) = 0 \) inflection? Concave Up Concave Down Need More Info

First vs Second Derivative Test

First Derivative Test Second Derivative Test
Requires Sign chart of \( f' \) around \( c \) Single evaluation of \( f''(c) \)
Works when Always (if \( f' \) is defined nearby) Only if \( f''(c) \neq 0 \)
Handles \( f'(c) \) DNE Yes No
Speed More steps Faster
Inconclusive case None When \( f''(c) = 0 \)

How to Apply the Test

1
Find \( f'(x) \). Differentiate the function.
2
Solve \( f'(x) = 0 \). Identify all critical points.
3
Find \( f''(x) \). Differentiate \( f' \) again.
4
Evaluate \( f''(c) \) at each critical point \( c \).
5
Classify: positive → local min, negative → local max, zero → inconclusive.

Example 1: \( f(x) = x^3 - 3x + 2 \)

1
\( f'(x) = 3x^2 - 3 \)
2
Set \( f'(x) = 0 \): \( 3x^2 - 3 = 0 \implies x^2 = 1 \implies x = \pm 1 \)
3
\( f''(x) = 6x \)
4
Evaluate: \( f''(-1) = -6 \) and \( f''(1) = 6 \)

Graph of \( f(x) = x^3 - 3x + 2 \)

Example 2: \( g(x) = x^4 - 4x^2 \)

1
\( g'(x) = 4x^3 - 8x = 4x(x^2 - 2) \)
2
Critical points: \( x = 0,\; x = \sqrt{2},\; x = -\sqrt{2} \)
3
\( g''(x) = 12x^2 - 8 \)
4
Evaluate: \( g''(0) = -8 \), \( g''(\sqrt{2}) = 16 \), \( g''(-\sqrt{2}) = 16 \)

Graph of \( g(x) = x^4 - 4x^2 \)

Example 3: Inconclusive Case

Consider \( h(x) = x^4 \). Find and classify all critical points.

1
\( h'(x) = 4x^3 \) → Critical point at \( x = 0 \).
2
\( h''(x) = 12x^2 \) → \( h''(0) = 0 \). Inconclusive!
3
First Derivative Test: \( h'(x) = 4x^3 \) changes from negative to positive at \( x = 0 \).
4
Conclusion: \( x = 0 \) is a local minimum despite \( h''(0) = 0 \).

Why \( f''(c) = 0 \) is Ambiguous

All three functions below have \( f'(0) = 0 \) and \( f''(0) = 0 \), yet each behaves differently at \( x = 0 \):

\( x^4 \)
Local minimum at \( x = 0 \).
\( -x^4 \)
Local maximum at \( x = 0 \).
\( x^3 \)
Neither: inflection point at \( x = 0 \).

Common Mistakes to Avoid

✕ Forgetting \( f'(c) = 0 \)
The test requires \( c \) to be a critical point. If \( f'(c) \neq 0 \), the Second Derivative Test does not apply.
✕ Claiming a Result at \( f''(c) = 0 \)
When \( f''(c) = 0 \), students often guess "inflection point." The test is inconclusive: you must verify with another method.
✕ Mixing Up the Sign
Positive \( f''(c) \) means concave up (min), not max. Remember: smile = min, frown = max.

Quick Reference Summary

Value of \( f''(c) \) Concavity Classification
\( f''(c) > 0 \) Concave Up Local Minimum
\( f''(c) < 0 \) Concave Down Local Maximum
\( f''(c) = 0 \) Unknown Inconclusive

Question 1 of 5

Let \( f(x) = x^3 - 12x + 5 \). Using the Second Derivative Test, classify the critical point at \( x = 2 \).

Question 2 of 5

For the function \( g(x) = -2x^2 + 8x - 3 \), the critical point is at \( x = 2 \). What does the Second Derivative Test tell us?

Question 3 of 5

Consider \( p(x) = x^5 \). The Second Derivative Test at \( x = 0 \) yields:

Question 4 of 5

If \( f'(3) = 0 \) and \( f''(3) = 7 \), which statement is correct?

Question 5 of 5

For \( f(x) = \sin x \) on the interval \( (0, 2\pi) \), which critical point is a local maximum?