Concave down

Definition

A function is concave down on an interval if its second derivative is negative on that interval. This means the graph of the function bends downward like an upside-down cup.

5 Must Know Facts For Your Next Test

1. If $f''(x) < 0$ for all $x$ in an interval, then the function is concave down on that interval.
2. Concavity can change at points where the second derivative is zero or undefined, known as inflection points.
3. A concave down function suggests that the rate of change of the slope (first derivative) is decreasing.
4. The tangent line to a curve at any point where it is concave down lies above the curve near that point.
5. Understanding concavity helps in sketching more accurate graphs and in determining local maxima.

Review Questions

• What does it mean for a function to be concave down?
• How do you determine if a function is concave down using its second derivative?
• What are inflection points, and how do they relate to concavity?

Related terms

Second Derivative: The derivative of the first derivative of a function, denoted as $f''(x)$. It provides information about the curvature of the graph.

Concave Up: A function is concave up on an interval if its second derivative is positive on that interval, making the graph bend upward.

Inflection Point: A point on a curve where the sign of the curvature (concavity) changes. It occurs where $f''(x)$ changes from positive to negative or vice versa.