Is a point of inflection concave or convex? - Which answer is correct?

In summary, a point of inflection is a point on a graph where the concavity changes, meaning the slope of the graph transitions from increasing to decreasing or vice versa. It can be either concave or convex, depending on the shape of the graph before and after the point. The concavity can be determined by looking at the direction of the curve before and after the point. Both concave and convex answers can be correct, as a point of inflection can be either one. It cannot be both concave and convex at the same time.
  • #1
Sarah66
1
0
Homework Statement
Hello everybody, :)
I have a problem with determining the intervals of convexity and concavity of this function:
Relevant Equations
there are no inflection points here for me
problem1.png

but for me there is no solution to this inequality...
I never wait for a ready answer but I've already spent 4 hours on it and I still don't know what to mark...
 
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  • #2
Sarah66 said:
there are no inflection points here for me
There doesn’t need to be an inflection point to switch between concave and convex if there are points where the function is undefined.
 
  • #3
Check the sign of the 2nd derivative on each interval if you aren't sure.
 

1. Is a point of inflection always concave?

No, a point of inflection can be either concave or convex depending on the direction of the curve at that point.

2. How can I determine if a point of inflection is concave or convex?

You can determine the direction of the curve at a point of inflection by looking at the concavity of the curve on either side of the point. If the curve is concave up on one side and concave down on the other, the point of inflection is concave. If the curve is concave down on both sides or concave up on both sides, the point of inflection is convex.

3. Can a point of inflection be both concave and convex?

No, a point of inflection can only be either concave or convex. It cannot be both at the same time.

4. Why is it important to know if a point of inflection is concave or convex?

Knowing the direction of a point of inflection can help in analyzing the behavior of a curve and understanding its properties. It can also be useful in determining the maximum or minimum points of a curve.

5. Which answer is correct - concave or convex?

Both answers can be correct depending on the direction of the curve at the point of inflection. Therefore, it is important to specify the direction when discussing a point of inflection.

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