Definitely Maybe: Concavity and Point of Inflection

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In summary, the given information provides a Second Derivative Sign Test to determine the existence of a point of inflection at x = 5 for a continuous function f on the interval [0, 8]. The test shows a sign change at f''(5), indicating a possible point of inflection. However, due to the lack of derivative at x = 5, it is uncertain whether there is a tangent line and thus, the answer could either be "definitely" or "possibly." Two examples are provided to illustrate this uncertainty.
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Homework Statement


f is a continuous function on [0, 8] and satisfies the following:
Second Derivative Sign Test
x ; f''
0 [tex]\leq[/tex] x < 3 ; -
3 ; 0
3 < x < 5 ; +
5 ; Does Not Exist
5 < x < 6 ; -
6 ; 0
6 < x [tex]\leq[/tex] 8 ; -

Based on this information is there a point of inflection at x = 5?
(a) Definitely
(b) Possibly
(c) Definitely not

Homework Equations


Definition: A point of inflection is where the concavity changes and there is a tangent line.

Concavity changes when f'' changes signs. Negative f'' means concave down; positive f'' means concave up.

The Attempt at a Solution


There is a sign change at f''(5), but the correct answer could either be "definitely" or "possibly." The point of inflection does not exist, but there is still a sign change, meaning it is a point of inflection. But my class and I were wondering whether it was possible that because there is no derivative at x = 5, there could also possibly be or not be a point of inflection there, too. The book says it's "possibly" a point of inflection, but going by the definition, we just presumed it to "definitely" be one.
 
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dozzer said:
Definition: A point of inflection is where the concavity changes and there is a tangent line.

Can you think how this would reflect on the question? Can you think of two examples to satisfy the conditions on f''(x), one that does not have a tangent at x=5 and one which does?
 

Related to Definitely Maybe: Concavity and Point of Inflection

1. What is concavity?

Concavity refers to the curvature or shape of a graph. It can be either concave up, where the graph is curving upwards, or concave down, where the graph is curving downwards.

2. What is a point of inflection?

A point of inflection is a point on a graph where the concavity changes. This means that the graph goes from being concave up to concave down, or vice versa, at that specific point.

3. How do you determine the concavity of a function?

To determine the concavity of a function, you can take the second derivative of the function. If the second derivative is positive, the graph is concave up. If the second derivative is negative, the graph is concave down.

4. Can a function have more than one point of inflection?

Yes, a function can have multiple points of inflection. This occurs when the concavity changes more than once on the graph.

5. How can understanding concavity and points of inflection be useful in real life?

Understanding concavity and points of inflection can be useful in fields such as economics, engineering, and physics. It can help determine optimal solutions for problems involving rates of change, as well as identifying critical points in a system.

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