2.1

Concavity and the Second Derivative

Understanding Concavity

Concavity describes the direction in which a function curves. It is closely related to the behavior of the function's slope and is determined using the second derivative.

• Concave Up: A function f is concave up on an interval if its graph lies above its tangent lines on that interval. This means the slope of the function is increasing as you move from left to right.

• Concave Down: A function f is concave down on an interval if its graph lies below its tangent lines on that interval. This means the slope of the function is decreasing as you move from left to right.

Mathematical Criteria:

• If $f''(x) > 0$ on an interval, then f is concave up on that interval.

• If $f''(x) < 0$ on an interval, then f is concave down on that interval.

Example: The graph of $f(x) = x^2$ is concave up everywhere because $f''(x) = 2 > 0$ for all $x$.

Graphical Interpretation

• When the slope (first derivative) is increasing, the graph is concave up (shaped like a cup).

• When the slope is decreasing, the graph is concave down (shaped like a cap).

• Points where the concavity changes are called inflection points.

Inflection Points

Definition and Identification

An inflection point is a point on the graph of a function where the concavity changes from up to down or vice versa. At an inflection point, the second derivative is typically zero or undefined, and the function must be continuous at that point.

• To find inflection points:

  1. Find where $f''(x) = 0$ or $f''(x)$ is undefined.

  2. Check for a change in sign of $f''(x)$ around those points.

Example: For $f(x) = x^3$, $f''(x) = 6x$. At $x = 0$, $f''(x)$ changes sign, so $(0,0)$ is an inflection point.

Relative and Absolute Extrema

Definitions

• Relative (Local) Maximum: A point where the function value is higher than all nearby points.

• Relative (Local) Minimum: A point where the function value is lower than all nearby points.

• Absolute Maximum: The highest value of the function on its entire domain.

• Absolute Minimum: The lowest value of the function on its entire domain.

First Derivative Test: Extrema occur where $f'(x) = 0$ or $f'(x)$ is undefined. The sign of $f'(x)$ changes at these points.

Second Derivative Test:

• If $f'(c) = 0$ and $f''(c) > 0$, then $f$ has a local minimum at $x = c$.

• If $f'(c) = 0$ and $f''(c) < 0$, then $f$ has a local maximum at $x = c$.

Intervals of Increase and Decrease

Determining Intervals

A function is increasing where its first derivative is positive and decreasing where its first derivative is negative.

• Increasing: $f'(x) > 0$

• Decreasing: $f'(x) < 0$

Example: If $f'(x) > 0$ for $x$ in $(a, b)$, then $f$ is increasing on $(a, b)$.

Worked Example: Analyzing a Graphed Function

Identifying Extrema and Concavity

Given a graphed function, you can identify key features as follows:

• Absolute Minimum: The lowest point on the graph.

• Relative Maxima: Points higher than their immediate neighbors.

• Relative Minima: Points lower than their immediate neighbors.

• Intervals of Increase: Where the graph rises as you move left to right.

• Intervals of Decrease: Where the graph falls as you move left to right.

• Intervals of Concave Up: Where the graph is shaped like a cup.

• Intervals of Concave Down: Where the graph is shaped like a cap.

Feature	Location (x-value)	y-value
Absolute Minimum	None (in this example)	—
Relative Maximum	x = -1	4
Relative Minimum	x = 6.4	-8
Relative Minimum	x = 1.8	-3.8

• Function increases on intervals: $(-3, -1) \cup (1.8, 6.4)$

• Function decreases on intervals: $(-\infty, -3) \cup (-1, 1.8) \cup (6.4, \infty)$

• Function is concave up on intervals: $(-\infty, -1) \cup (-1, 4)$

• Function is concave down on intervals: $(4, \infty)$

Additional info: In business calculus, understanding concavity and extrema is essential for analyzing cost, revenue, and profit functions, as well as for optimizing business decisions.