Applications of the Derivative

First Derivative Test

The First Derivative Test is used to determine the relative (local) maxima and minima of a function by analyzing the sign changes of its first derivative at critical points.

• Critical Points: Points where or is undefined.

• Relative Maximum: If changes from positive to negative at , then has a local maximum at $x = c$.

• Relative Minimum: If changes from negative to positive at , then has a local minimum at $x = c$.

• Example: For , the critical point is . Analyze around $x = 1$ to determine if it is a max or min.

Second Derivative Test

The Second Derivative Test provides another method to classify critical points using the value of the second derivative at those points.

• Test Conditions:

  • If and , then has a local minimum at .

  • If and , then has a local maximum at .

  • If and , the test is inconclusive.

• Example: For , find critical points by solving , then use to classify each.

Concavity and Inflection Points

Concavity describes the direction in which a function curves. The second derivative indicates concavity:

• Concave Up: If on an interval, the graph is concave up (shaped like a cup).

• Concave Down: If on an interval, the graph is concave down (shaped like a cap).

• Inflection Point: A point where the concavity changes (i.e., changes sign).

• Example: For , . At , changes sign, so $x = 0$ is an inflection point.

Examples and f. Assumptions about the Problem

• Example 1: . At , , but the sign does not change, so $x = 0$ is not an inflection point.

• Example 2: . The sign of changes at , so $x = -2$ is an inflection point. The function is concave down for and concave up for .

• Example 3: . Since is always non-negative and only zero at , but the sign does not change, there are no inflection points. The function is concave up everywhere except at .

Summary Table: Second Derivative Test

Summary Table: Concavity and Inflection Points

Key Definitions

• Critical Point: Where or is undefined.

• Inflection Point: Where and concavity changes.

• Concave Up: .

• Concave Down: .

Additional info: Some examples and explanations were expanded for clarity and completeness, including step-by-step analysis of concavity and inflection points.