Concavity and the Second-Derivative Test

Concavity

• Graphical Interpretation: A curve is concave upward if it lies above its tangent lines and concave downward if it lies below its tangent lines.

• Test for Concavity: Let $f$ be a function whose second derivative exists on an open interval.

  • If f''(x) > 0 for all $x$ in the interval, the graph of $f$ is concave upward.

  • If f''(x) < 0 for all $x$ in the interval, the graph of $f$ is concave downward.

Points of Inflection

• Definition: A point of inflection is a point on the graph where the concavity changes, provided a tangent line exists at that point.

• Locating Points: Possible points of inflection occur at values of $x$ for which $f''(x) = 0$ or for which $f''(x)$ is undefined.

• Requirement for Inflection: The sign of $f''(x)$ must change as you move across the point.

The Second-Derivative Test

• The test uses the second derivative at a critical point where $f'(c) = 0$ to determine relative extrema:

  1. If f''(c) > 0, then $f(c)$ is a relative minimum.

  2. If f''(c) < 0, then $f(c)$ is a relative maximum.

  3. If $f''(c) = 0$, the test fails. The First-Derivative Test must be used instead.

Extended Application: Diminishing Returns

• Input-Output Models: The point of diminishing returns is the inflection point $(c, f(c))$ where the graph transitions from concave upward to concave downward.

• Economic Interpretation:

  • On the concave upward interval, each additional dollar of input returns more than the previous dollar.

  • On the concave downward interval, each additional dollar of input returns less than the previous dollar.