Lesson 4: 5.4 The Second Derivative Test
Duration of Days: 1
Lesson Objective
Apply the Second Derivative Test to classify critical points as relative maxima or relative minima.
Understand the conditions under which the Second Derivative Test is inconclusive.
Compare and contrast the First Derivative Test and the Second Derivative Test to determine which is more efficient for a given problem.
Relate the concavity of a function at a critical point to its status as a peak or a valley.
If the slope is zero and the graph is "bending upward," why must that point be a minimum?
Why is the Second Derivative Test often faster than a sign chart when dealing with trigonometric or transcendental functions?
What do we do if the Second Derivative Test fails to give us a definitive answer?
Second Derivative Test
Concavity
Relative Maximum
Relative MinimumIn
conclusive f''(x) Notation
Standard FUN-1.C: Justify the classification of a relative extremum using the Second Derivative Test.
Mathematical Practice 3.D: Provide reasons or rationales for conclusions.
Description:
While the First Derivative Test looks at the change in sign of f', the Second Derivative Test looks at the value of f'' at a specific point.
Find where f'(c) = 0.
Find f''(x).
Plug c into f''(x).
If f''(c) > 0, the graph is concave up (a "cup"), making c a relative minimum.
If f''(c) < 0, the graph is concave down (a "frown"), making c a relative maximum.
Purpose: This test is a frequent requirement on the AP Exam, especially in problems where the first derivative is given as an expression that is difficult to use in a sign chart. It reinforces the conceptual link between concavity and extrema.
DOK 1 (Recall): Stating the conditions of the Second Derivative Test.
DOK 2 (Skill/Concept): Computing the second derivative and evaluating it at critical points to classify extrema.
DOK 3 (Strategic Thinking): Justifying why the Second Derivative Test is inconclusive for a function like f(x) = x^4 at x=0, and switching to the First Derivative Test to finish the analysis.
Scaffolding (Support):The "Smiley/Frowny" Mnemonic: * Positive Second Derivative = Smile (Minimum at the bottom).
Negative Second Derivative (-) = Frown (Maximum at the top).
Comparison Graphic: Create a T-chart showing when to use the 1st vs. 2nd Derivative Test.
Extension (Challenge):Unknown Functions: Provide a table of values for $, f', f'' at various x-values.
Ask students to identify all relative extrema without ever seeing the actual equation of the function.
Parameters: Solve for constants a and b in f(x) = ax^3 + bx^2 such that the function has a relative maximum at a specific coordinate.
AP College Board Assessments