Second Derivative

4.2 Applications of the Second Derivative

Concavity of Functions

  • Definition: Let the function (f) be differentiable on an interval ((a, b)).

    • Intuitive Explanation:

    • The graph of (f) is concave up on ((a, b)) if its first derivative (f'(x)) is increasing on ((a, b)).

    • The graph of (f) is concave down on ((a, b)) if its first derivative (f'(x)) is decreasing on ((a, b)).

    • Mathematical Criteria:

    • The graph of (f) is concave up on ((a, b)) if (f''(x) > 0) for all (x) in ((a, b)).

    • The graph of (f) is concave down on ((a, b)) if (f''(x) < 0) for all (x) in ((a, b)).

Theorems Regarding Concavity

  • Theorem 2:

    • a. If (f''(x) > 0) for all (x) in ((a, b)), then the graph of (f) is concave up on ((a, b)).

    • b. If (f''(x) < 0) for all (x) in ((a, b)), then the graph of (f) is concave down on ((a, b)).

  • Summary: The second derivative (f''(x)) determines whether (f(x)) is concave up or concave down.

Determining the Intervals of Concavity of the Graph of (f)

  1. Find all values of (x) for which (f''(x) = 0) or (f''(x) \text{ is undefined}). These values define points where (f''(x)) changes sign.

  2. Perform a number line test:

    • Divide the real number line using the values found in Step 1.

    • Select a convenient test point in each interval and evaluate (f''(c)).

    • a. If (f''(c) > 0), then the graph of (f) is concave up on that interval.

    • b. If (f''(c) < 0), then the graph of (f) is concave down on that interval.

Inflection Points

  • Definition: An inflection point is a point on the graph of a continuous function (f) where the tangent line exists and where the concavity changes.

Finding Inflection Points of a Continuous Function

  1. Compute the second derivative (f''(x)).

  2. Determine the numbers in the domain of (f) for which (f''(x) = 0) or (f''(x) \text{ is undefined}).

  3. Determine the sign of (f''(x)) to the left and right of each number (c) found in Step 2. If there is a change in the sign of (f''(x)) across (x = c), then ((c, f(c))) is an inflection point of (f).

Example 1: Finding Concavity and Inflection Points

  • Function: Let (g(x) = x^4 - 2x^3 + 6).

    • (a) Find the interval(s) where (g(x)) is concave up and the intervals where (g(x)) is concave down.

    • (b) Find the inflection point(s) of (g(x)).

The Second Derivative Test

  • Definition: The Second Derivative Test is an alternative to the First Derivative Test (section 4.1) for classifying the critical numbers of (f) as relative maxima or relative minima.

Steps for the Second Derivative Test

  1. Compute the first and second derivatives (f'(x)) and (f''(x)).

  2. Find all critical numbers of (f) (i.e. numbers in the domain of (f) where (f'(x) = 0) or (f'(x) \text{ is undefined})).

  3. Compute (f''(c)) for each critical number (c):

    • a. If (f''(c) < 0), then (f) has a relative maximum at (c).

    • b. If (f''(c) > 0), then (f) has a relative minimum at (c).

    • c. If (f''(c) = 0) or (f''(c) \text{ does not exist}), the test fails, indicating that the result is inconclusive.

Example 2: Using the Second Derivative Test

  • Function: Consider the function (f(x) = -x^3 + 6x^2).

    • Use the Second Derivative Test to find the relative maxima and minima of (f(x)).

Supplemental Examples

  • Example 3: Determine where the graph of the function (f(x) = \frac{x + 1}{x - 1}) is concave up and where it is concave down.

Course: MATH 221 - University of Delaware