Concavity test
Definition
The concavity test determines where a function is concave up or concave down by analyzing the sign of its second derivative. If $f''(x) > 0$, the function is concave up at that point, and if $f''(x) < 0$, it is concave down.
5 Must Know Facts For Your Next Test
- The second derivative test can also identify points of inflection where concavity changes.
- A function is concave up on an interval if $f''(x) > 0$ for all $x$ in that interval.
- A function is concave down on an interval if $f''(x) < 0$ for all $x$ in that interval.
- At a point of inflection, the second derivative either changes sign or is zero.
- Concavity provides insight into the shape and curvature of a graph, helping to understand behavior such as acceleration and deceleration.
Review Questions
- What does it mean for a function to be concave up?
- How do you determine intervals of concavity using the second derivative?
- What information can be gained from identifying points of inflection?
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Related terms
Second Derivative: The derivative of the first derivative, used to analyze the concavity and inflection points of a function.
Point of Inflection: A point where a function changes its concavity, indicated by a change in sign of the second derivative.
First Derivative Test: A method used to find local maxima and minima by analyzing the sign changes in the first derivative.
