AP Calculus AB Flashcards: Second Derivative Test

What this deck covers

This deck focuses on Second Derivative Test, giving you a quick way to review the definitions, rules, and examples that matter most for AP Calculus AB.

How to use these flashcards

Work through these flashcards in short sessions. Try to answer each prompt before flipping the card, then revisit any cards you miss until the explanation feels automatic.

Flashcard 1: For $f(x) = x^3 - 6x^2 + 12x - 5$, determine $f''(x)$.

Answer: $f''(x) = 6x - 12$.. Second derivative of $x^3 - 6x^2 + 12x - 5$ using power rule.

Flashcard 2: What does $f''(x) < 0$ indicate about the interval?

Answer: The function is concave down on the interval. Negative second derivative means downward curvature.

Flashcard 3: Identify the extremum type if $f''(c) = 0$ at $c$.

Answer: Inconclusive. Zero second derivative provides no classification information.

Flashcard 4: What is the term for $f''(c) < 0$ at a critical point $c$?

Answer: Local maximum. Negative second derivative at a critical point indicates a maximum.

Flashcard 5: What does $f''(x) > 0$ indicate about the interval?

Answer: The function is concave up on the interval. Positive second derivative means upward curvature.

Flashcard 6: Determine $f''(x)$ for $f(x) = 6x^3 - 3x^2 + 2$.

Answer: $f''(x) = 36x - 6$. Second derivative of $6x^3 - 3x^2 + 2$ using power rule.

Flashcard 7: For $f(x) = x^4 - 4x^2$, determine $f''(x)$.

Answer: $f''(x) = 12x^2 - 8$. Second derivative of $x^4 - 4x^2$ using power rule.

Flashcard 8: If $f''(x) < 0$ for all $x$ in an interval, what can be concluded about $f(x)$?

Answer: $f(x)$ is concave down on that interval. Negative second derivative means the graph curves downward.

Flashcard 9: Calculate $f''(3)$ for $f(x) = x^3 - 6x^2 + 9x + 1$.

Answer: $f''(3) = 12$. Substituting $x = 3$ into $f''(x) = 6x - 12$.

Flashcard 10: State the condition for a local maximum using the Second Derivative Test.

Answer: If $f''(c) < 0$, $f(c)$ is a local maximum. Negative second derivative indicates concave down, creating a maximum.

Flashcard 11: Find $f''(x)$ for $f(x) = 2x^2 - 4x + 1$.

Answer: $f''(x) = 4$. Second derivative of quadratic function is constant.

Flashcard 12: State the condition for a local minimum using the Second Derivative Test.

Answer: If $f''(c) > 0$, $f(c)$ is a local minimum. Positive second derivative indicates concave up, creating a minimum.

Flashcard 13: Identify the first step in applying the Second Derivative Test.

Answer: Find the critical points where $f'(x) = 0$ or $f'(x)$ is undefined. Critical points are necessary candidates for local extrema.

Flashcard 14: What is the relationship between concavity and the second derivative?

Answer: Concavity is determined by the sign of the second derivative. Second derivative sign determines upward or downward curvature.

Flashcard 15: What is the geometrical interpretation of a local maximum?

Answer: A point where the curve changes from increasing to decreasing. The highest point in a local neighborhood of the function.

Flashcard 16: Determine the concavity of $f(x)$ if $f''(x) = 0$ for all $x$ in an interval.

Answer: The concavity cannot be determined. Zero second derivative gives no concavity information.

Flashcard 17: If $f''(x) = 15x - 5$, find the concavity at $x = 1$.

Answer: Concave up since $f''(1) = 10 > 0$. Substituting $x = 1$ gives $f''(1) = 15(1) - 5 = 10 > 0$.

Flashcard 18: What does the Second Derivative Test determine at $c$?

Answer: Whether $f(c)$ is a local max, min, or inconclusive. Classifies critical points as maxima, minima, or undetermined.

Flashcard 19: Determine the extremum type for $f(x) = x^2 - 4x$ at $x = 2$.

Answer: Local minimum since $f''(2) > 0$. Since $f''(2) = 2 > 0$, the critical point is a minimum.

Flashcard 20: Find the local extremum for $f(x) = x^3 - 3x$ at $x = 1$.

Answer: Local minimum since $f''(1) = 6 > 0$. Since $f''(1) = 6 > 0$, the critical point is a minimum.

Flashcard 21: When is the Second Derivative Test inconclusive?

Answer: When $f''(c) = 0$ at a critical point $c$. Zero second derivative at critical points gives no information.

Flashcard 22: Find $f''(x)$ for $f(x) = 3x^3 - 9x^2$.

Answer: $f''(x) = 18x - 18$. Second derivative of $3x^3 - 9x^2$ using power rule.

Flashcard 23: Find the second derivative of $f(x) = x^5 - 5x^4$.

Answer: $f''(x) = 20x^3 - 60x^2$. Taking derivative twice of $x^5 - 5x^4$ using power rule.

Flashcard 24: What is the geometrical interpretation of a local minimum?

Answer: A point where the curve changes from decreasing to increasing. The lowest point in a local neighborhood of the function.

Flashcard 25: Calculate $f''(2)$ for $f(x) = x^3 - 6x^2 + 9x + 1$.

Answer: $f''(2) = 6$. Substituting $x = 2$ into $f''(x) = 6x - 12$.

Flashcard 26: Find $f''(x)$ for $f(x) = x^3 - 3x + 1$.

Answer: $f''(x) = 6x$. Second derivative of $x^3 - 3x + 1$ using power rule.

Flashcard 27: What is concavity in the context of calculus?

Answer: The direction of the curve of a function. Describes whether a graph curves upward or downward.

Flashcard 28: If $f''(x) > 0$ for all $x$ in an interval, what can be concluded about $f(x)$?

Answer: $f(x)$ is concave up on that interval. Positive second derivative means the graph curves upward.

Flashcard 29: What is a critical point in the context of the Second Derivative Test?

Answer: A point where $f'(x) = 0$ or $f'(x)$ is undefined. Where the first derivative equals zero or doesn't exist.

Flashcard 30: What does $f''(c) = 0$ imply in the Second Derivative Test?

Answer: The test is inconclusive. Zero second derivative provides no information about extremum type.