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- For example, when a tangent drawn at a point lies below the graph in the vicinity of that point, the graph is said to be concave up. Conversely, when a tangent drawn at a point lies above the graph in the vicinity of that point, the graph is said to be concave downward.
testbook.com/maths/concave-functionConcave Functions: Definition, Properties with Solved Examples
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Dec 21, 2020 · 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. We have been learning how the first and second derivatives of a function relate information about the graph of that function.
Example 1: Characterising Graphs. Say we have a graph of the function f(x) = x(x^2 + 1). Find the parts of the graph where the function is convex or concave, and find the point(s) of inflexion. [3 marks] f(x) = x(x^2 + 1) = x^3 + x gives. f''(x) = 6x. f''(x) = 0, when x = 0. f''(x) \textcolor{red}{< 0} when x<0. Here we have a concave section.
A point \( P \) on the graph of \( y = f(x) \) is a point of inflection if \( f \) is continuous at \( P \) and the concavity of the graph changes at \( P \). In view of the above theorem, there is a point of inflection whenever the second derivative changes sign.
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 · A point \(x = c\) is called an inflection point if the function is continuous at the point and the concavity of the graph changes at that point. Now that we have all the concavity definitions out of the way we need to bring the second derivative into the mix.
The concavity of the graph of a function refers to the curvature of the graph over an interval; this curvature is described as being concave up or concave down. Generally, a concave up curve has a shape resembling "∪" and a concave down curve has a shape resembling "∩" as shown in the figure below. Concave up.
Graphically, a function is concave up if its graph is curved with the opening upward (Figure \(\PageIndex{1a}\)). Similarly, a function is concave down if its graph opens downward (Figure \(\PageIndex{1b}\)). Figure \(\PageIndex{1}\) This figure shows the concavity of a function at several points.
