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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.
If \( f''(x) \) is positive on an interval, the graph of \( y=f(x) \) is concave up on that interval. We can say that \(f\) is increasing (or decreasing) at an increasing rate. If \( f''(x) \) is negative on an interval, the graph of \( y=f(x) \) is concave down on that interval.
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 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.
Figure 1. This figure shows the concavity of a function at several points. Notice that a function can be concave up regardless of whether it is increasing or decreasing.
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Identifying concavity and points of inflections given a function’s graph. Applying the second derivative test to determine the concavity of a function at different critical points. Understanding the relationship between f (x), f ′ (x), and f ′ ′ (x) and how it affects the function’s shape.
