Concavity and Second Derivative Notes

Concavity and the Second Derivative Notes

Definition of Concavity

• Let f be differentiable on an open interval I.

• The graph of f is concave upward on I when:

  • The first derivative f' is increasing on the interval.

• The graph of f is concave downward on I when:

  • The first derivative f' is decreasing on the interval.

Graphical Interpretation of Concavity

1. Concave Upward:

   • If the graph of f is concave upward on I, then:

     • The graph of f lies above all of its tangent lines on I.

     • Visual Reference: See Figure 3.23(a).

2. Concave Downward:

   • If the graph of f is concave downward on I, then:

     • The graph of f lies below all of its tangent lines on I.

     • Visual Reference: See Figure 3.23(b).

Determining Concavity Analytically

• To ascertain the open intervals on which the graph of a function f is concave upward or downward, it is crucial to identify the intervals on which f' is increasing or decreasing.

• Example: Given the function $f(x) = rac{1}{3} x^3 - x$

  • Determine the intervals where f(x) is concave up or concave down.

Determining Concavity Graphically

• To use Theorem 3.7:

  1. Locate the x-values at which the second derivative f"(x) = 0 or where it does not exist.

  2. Use these x-values to establish test intervals.

  3. Test the sign of f"(x) in each of the test intervals.

  • Example: Analyze the graph of
    $f(x) = 6x^2 + 3$

  • Determine the open intervals of concavity.

Continuity and Test Intervals

• The function examined previously is continuous across the entire real number line.

• In cases where there are x-values where the function is not continuous, these values, as well as points where f"(x) = 0 or f"(x) does not exist, must be utilized to establish test intervals.

Points of Inflection

• A point of inflection occurs when:

  • The tangent line to the graph exists at a point where the concavity changes.

  • At a point of inflection, the graph crosses its tangent line.

• Three types of points of inflection illustrate:

  • The concavity of f changes at these points.

  • Note: Points of inflection are at the critical numbers of f’.

• Exercise: Determine the points of inflection and discuss the concavity of the graph for the function
  $f(x) = x^4 - 4x^3$.

The Second Derivative Test

• The converse of Theorem 3.8 does not hold universally.

• It is possible for the second derivative to equal 0 at a point that is not a point of inflection.

Exercises and Homework

• Task: Find the relative extrema of the function
  $f(x) = -3x^5 + 5x^3$.

• Complete all assigned problems in your bound notebook and ensure to show all required work.

• Communication of solutions, thoughts, and processes is integral.

• Labeling: Clearly label each page/source and each problem with its specific number.

• Textbook Problems:

  • Problems from page 196: 7, 9, 13, 17, 20, 22, 24, 25, 27, 31, 33-43 (odd), 47, 51, 53, 55.