Concavity; Second Derivative Test

Definition

• The graph of f(x) is concave upward if f '(x) is increasing on the interval (or the graph of f(x) lies above its tangent lines).
• The graph of f(x) is concave downward if f '(x) is decreasing on the interval (or the graph of f lies below its tangent lines).

Concavity Test

• If f "(x) > 0, then f(x) is concave up.
• If f "(x) < 0, then f(x) is concave down.

Definition

• If f(x) has a tangent line at (c, f(c)), the point is called a point of inflection if the concavity of f(c) changes from upward to downward (or visa versa) at that point.

Second Derivative Test

• Given:  f '(c) = 0 and f "(c) exists
• Then:
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, then the test fails


Proof:

1. Suppose f "(c) > 0.
2. Then f(x) is concave up and lies above its tangent lines.
3. Since f '(c) = 0, the tangent line is horizontal at (c, f(c)).
4. Therefore, f(c) is a minimum.