Second Derivative Test
The second derivative test is a method to determine the concavity of a function and locate relative extrema. According to second derivative test:
- If f ‘′(c) > 0, then the function has a local minimum at x=c.
- If f ′′(c) < 0, then the function has a local maximum at x=c.
- If f ′′(c) = 0, the test is inconclusive.
Concavity and Points of Inflection
Concavity and points of inflection are the key concepts and basic fundamentals of calculus and mathematical analysis. It provides an insight into how curves behave and the shape of the functions. Where concavity helps us to understand the curving of a function, determining whether it is concave upward or downward, the point of inflection determines the point where the concavity changes, i.e., where either curve transforms from concave upward to concave downward or concave to convex, and vice versa. These concepts are essential in various mathematical applications, including curve sketching, optimization problems , and the study of differential equations.
In this article, we’ll shed lights on the definitions, properties, and practical implications of concavity and points of inflection.
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