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When f''(x) \textcolor{red}{< 0}, we have a portion of the graph where the gradient is decreasing, so the graph is concave at this section. An easy way to test for both is to connect two points on the curve with a straight line. If the line is above the curve, the graph is convex. If the line is below the curve, the graph is concave.
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.
- Definition of Concavity
- Theorem
- Definition of Point of Inflection
Let f′f′ be the first derivative of function ff that is differentiable on a given interval II, the graph of ff is (i) concave up on the interval II, if f′f′ is increasing on II, or (ii) concave down on the interval II, if f′f′ is decreasing on II. The sign of the second derivative informs us when f′f′is increasing or decreasing.
Let f″f′′ be the second derivative of function ff on a given interval II, the graph of ff is (i) concave up on II if f″(x)>0f′′(x)>0 on the interval II. (ii) concave down on II if f″(x)<0f′′(x)<0 on the interval II.
A point PP on the graph of y=f(x)y=f(x) is a point of inflection if ff is continuous at PP and the concavity of the graph changes at PP. In view of the above theorem, there is a point of inflection whenever the second derivative changes sign.
Definition. A function is concave up if the rate of change is increasing. A function is concave down if the rate of change is decreasing. A point where a function changes from concave up to concave down or vice versa is called an inflection point.
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.
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.
Of particular interest are points at which the concavity changes from up to down or down to up; such points are called inflection points. If the concavity changes from up to down at x = a, f ″ changes from positive to the left of a to negative to the right of a, and usually f ″ (a) = 0.
