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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.
- 3.4: Concavity and the Second Derivative
When the graph is concave up, the critical point represents...
- 3.4: Concavity and the Second Derivative
A point of inflexion occurs when the curve transitions from convex to concave or vice versa. We’re looking for sections of the graph where f'' (x) = 0. Note: While all points of inflexion have f'' (x) = 0, not all points where f'' (x) = 0 are points of inflexion.
- 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. Example 1: Describe the Concavity. An object is thrown from the top of a building.
′′If 𝑓(𝑥)<0 for all 𝑥 in 𝐼, then the graph of 𝑓 is concave down on 𝐼. Let 𝑓 be a function which is continuous at 𝑐 and differentiable near 𝑐. The point (𝑐,𝑓(𝑐)) is a point of inflection if the graph of 𝑓 changes concavity at 𝑥=𝑐. How do we find the points of inflection?
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.
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Concavity. 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. How to find the concavity of a function
