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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
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.
- 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.
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.
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. If f ″ (x) is positive on an interval, the graph of y = f(x) is concave up on that interval.
Lesson 7: Determining concavity of intervals and finding points of inflection: algebraic. Analyzing concavity (algebraic) Inflection points (algebraic) Mistakes when finding inflection points: second derivative undefined.
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point on a graph where the concavity of the curve changes (from concave down to concave up, or vice versa) is called a point of inflection (Definition 4.14). By implication (think about what separates positive and negative numbers on a number line), if a point (c, f (c)) is a point of inflection, then f ′ ′ ( c ) = 0 .
