Graph of Integral: Find x-coords of Points of Inflection

In summary, the key to solving this problem lies in understanding derivatives and the 2nd Fundamental Theorem.
  • #1
joess
16
0

Homework Statement


2ym93t2.jpg

Find the x-coordinate of each point of inflection of the graph of g on the open interval (-2, 5). Justify your answer

Homework Equations





The Attempt at a Solution


The answer (I think) is x=0 and x=3. Well, the thing is that I asked my tutor how to justify it and he said that it is only x=0 and not 3. But when I asked my teacher for the answer he checked and said it x=0, 3, but he couldn't tell me how to justify it. So now I'm wondering if I'm right or my tutor is.

The reason why I thought it was x=0 and x=3 is because from -2, the area as you move towards 0 is increasing at a greater and greater rate, therefore it's concave up. Then after 0 it's still increasing, but its rate of increase is decreasing, therefore concave down. Then at 2 it starts decreasing but the magnitude of the rate of decrease is still increasing (therefor still concave down) until x=3. Then after 3 it's still decreasing, but the magnitude of the rate is decreasing so it's concave up again. Then at 4 it starts to increase with increasing rate, and so g is still concave up.

But I'm pretty sure that on an exam they want a quick explanation in mathematical terms, not a whole long paragraph like that. My tutor said that because at x=3 there's a cusp so that's not a solution but I don't get why.:confused:
 
Physics news on Phys.org
  • #2
Your answers are correct - the key lies in understanding two things:

When do inflection points occur in a function?

How are the functions g and f related?

The answer to the first question (it has something to do with derivatives...) will help you answer the second one (along with the 2nd Fundamental Theorem).

Sorry to be so vague, but I want you to figure it out for yourself.
 
  • #3
Inflection points should occur where g''(x)=0...
And g'(x)=f(x), so g''(x)=f'(x), so they should occur where the slope changes from positive to negative or vice versa?
 
  • #4
What's your justification that g'(x) is differentiable? Remember, inflection points can happen where g''(x) doesn't exist.
 

Related to Graph of Integral: Find x-coords of Points of Inflection

1. What is a Point of Inflection in a graph of an integral?

A Point of Inflection in a graph of an integral is a point where the concavity of the graph changes. This means that the graph goes from being concave up to concave down, or vice versa. It is a point of change in the curvature of the graph.

2. How do I find the x-coordinates of Points of Inflection in a graph of an integral?

To find the x-coordinates of Points of Inflection in a graph of an integral, you first need to take the second derivative of the function. Then, set the second derivative equal to zero and solve for x. The resulting x-values are the x-coordinates of the Points of Inflection.

3. Can there be more than one Point of Inflection in a graph of an integral?

Yes, there can be more than one Point of Inflection in a graph of an integral. In fact, there can be an infinite number of Points of Inflection if the graph has a high enough degree or is a trigonometric function with multiple periods.

4. What is the significance of Points of Inflection in a graph of an integral?

Points of Inflection play an important role in understanding the shape and behavior of a graph of an integral. They can help determine the intervals where the function is increasing or decreasing, as well as the points where the graph changes direction.

5. How do Points of Inflection relate to the concept of concavity in a graph of an integral?

Points of Inflection and concavity are closely related in a graph of an integral. Points of Inflection mark the points where the concavity changes, and the concavity of the graph can be determined by analyzing the derivative of the function. A positive second derivative indicates a concave up graph, while a negative second derivative indicates a concave down graph.

Similar threads

Graphing Derivatives: Concavity and Inflection Points
    • Calculus and Beyond Homework Help
Replies
4
Views
1K
Inflection Point Calculation: Reduction of Cubic with Second Derivative Method
    • Calculus and Beyond Homework Help
Replies
1
Views
496
The integral of a function ##f(x)## from its graph
    • Calculus and Beyond Homework Help
Replies
15
Views
1K
Find the corresponding graphs for the distance-time graphs
    • Calculus and Beyond Homework Help
2
Replies
35
Views
2K
MHB Find where increasing/decreasing, concavity, local extrema and inflection points for f(x)=ln/x
    • Calculus
Replies
1
Views
1K
Graphing Derivatives: How to Find Maxima, Minima, and Points of Inflection
    • Calculus and Beyond Homework Help
Replies
7
Views
1K
The first and second derivatives at various points on a drawn graph
    • Calculus and Beyond Homework Help
Replies
22
Views
2K
Where are the extrema and inflection points for the function y = x/ln(x)?
    • Calculus and Beyond Homework Help
Replies
11
Views
1K
Finding x/y-intercepts, asymptotes, derivatives, and max min
    • Calculus and Beyond Homework Help
Replies
4
Views
1K
B Second derivatives and inflection points
    • Calculus
Replies
5
Views
2K

Hot Threads

Recent Insights

Back
Top