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Dec 21, 2020 · Figure \(\PageIndex{13}\): A graph of \(f(x)\) in Example \(\PageIndex{4}\). The second derivative is evaluated at each critical point. When the graph is concave up, the critical point represents a local minimum; when the graph is concave down, the critical point represents a local maximum.
Dec 21, 2020 · If we are trying to understand the shape of the graph of a function, knowing where it is concave up and concave down helps us to get a more accurate picture. Of particular interest are points at which the concavity changes from up to down or down to up; such points are called inflection points.
Nov 16, 2022 · The second derivative will allow us to determine where the graph of a function is concave up and concave down. The second derivative will also allow us to identify any inflection points (i.e. where concavity changes) that a function may have.
If f ′ (x) is positive on an interval, the graph of y = f(x) is increasing on that interval. If f ′ (x) is negative on an interval, the graph of y = f(x) is decreasing on that interval. The second derivative tells us if a function is concave up or concave down.
Concavity and convexity. It is said that a function f (x) is convex if, once having joined any two points of the graph, the segment stays over the graph: In this graph we can observe different segments (with different colors) that join two points of the graph and stay over it.
If a function changes from concave upward to concave downward or vice versa around a point, it is called a point of inflection of the function. In determining intervals where a function is concave upward or concave downward, you first find domain values where f″ (x) = 0 or f″ (x) does not exist.
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How to find concavity from the second derivative. To use the second derivative to find the concavity of a function, we first need to understand the relationships between the function f (x), the first derivative f' (x), and the second derivative f" (x).
