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
- If $f''(x) < 0$ for all $x$ in an interval, then the function is concave down on that interval.
- Concavity can change at points where the second derivative is zero or undefined, known as inflection points.
- A concave down function suggests that the rate of change of the slope (first derivative) is decreasing.
- The tangent line to a curve at any point where it is concave down lies above the curve near that point.
- 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?
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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.
