Applications of Differentiation

Using Second Derivatives to Find Maximum and Minimum Values and Sketch Graphs

This section explores how the second derivative of a function can be used to determine concavity, locate relative extrema (maximum and minimum points), and sketch the graph of a function. These concepts are fundamental in Business Calculus for analyzing and optimizing functions that model real-world scenarios.

Concavity of Functions

• Definition: The concavity of a function describes the direction in which the graph bends. If a function f is twice differentiable on an interval I:

  • Concave Up: f is concave up on I if its graph bends upwards, resembling a cup ().

  • Concave Down: f is concave down on I if its graph bends downwards, resembling a cap ().

• Visual Representation: Concave up graphs have slopes that increase, while concave down graphs have slopes that decrease.

Theorem 4: A Test for Concavity

• If on an interval I, the graph of f is concave up on I.

• If on an interval I, the graph of f is concave down on I.

• Application: This test helps identify regions where a function is increasing at an increasing rate (concave up) or increasing at a decreasing rate (concave down).

Theorem 5: The Second Derivative Test for Relative Extrema

• Suppose f is differentiable for every x in an open interval and there is a critical value in where .

• Relative Minimum: is a relative minimum if .

• Relative Maximum: is a relative maximum if .

• If , use the First Derivative Test to determine whether is a relative extremum.

Example 1: Finding Relative Extrema

Given the function , find its relative extrema.

• Step 1: Find the first and second derivatives.

• Step 2: Find critical values by solving .

  • Critical values: ,

• Step 3: Apply the Second Derivative Test.

  • (less than 0): Relative maximum at

  • (greater than 0): Relative minimum at

• Step 4: Find the function values at the critical points.

  • Conclusion: Relative maximum at and relative minimum at

Summary Table: Second Derivative Test

Key Terms

• Critical Value: A point where or does not exist.

• Relative Maximum: The highest point in a local region of the graph.

• Relative Minimum: The lowest point in a local region of the graph.

• Concavity: The direction in which the graph bends (upward or downward).

Example Application

• Business Context: In economics and business, finding maximum and minimum values helps identify optimal production levels, profit maximization, and cost minimization.

• Graphical Analysis: Sketching the graph using critical points and concavity provides insight into the behavior of functions modeling business scenarios.

Additional info: These notes are based on textbook slides for Chapter 3 of "Calculus and Its Applications," focusing on the use of second derivatives in curve sketching and optimization, which are core topics in Business Calculus.