Inflection point

Definition

An inflection point is a point on a curve where the concavity changes from concave up to concave down or vice versa. At this point, the second derivative of the function is zero or undefined.

5 Must Know Facts For Your Next Test

1. An inflection point occurs where the second derivative of a function changes sign.
2. To confirm an inflection point, both the second derivative test and a change in concavity must be checked.
3. If $f''(x) = 0$ or $f''(x)$ is undefined at $x = c$, it is a candidate for an inflection point.
4. Concavity shifts from upward (concave up) to downward (concave down) or vice versa at an inflection point.
5. Not every point where the second derivative is zero is necessarily an inflection point; it must also show a change in concavity.

Review Questions

• What conditions must be met for a point to be considered an inflection point?
• How do you determine if there is a change in concavity at a given point?
• Can a function have an inflection point where its second derivative is undefined? Explain.

Related terms

Second Derivative: The derivative of the first derivative of a function, often denoted as $f''(x)$, used to determine concavity and points of inflection.

Concavity:

The property of a curve describing whether it bends upwards (concave up) or downwards (concave down).

Critical Point: A point on the graph where the first derivative is zero or undefined. It can indicate local maxima, minima, or saddle points.