Finding Where F is concave up/down using the graph of the first derivative

In summary, to determine if a function is concave up or concave down using its first derivative graph, you need to look at the direction of the graph. A function cannot be both concave up and concave down at the same time, and can only have one type of concavity at a particular point. To tell if a function is concave up or concave down at a specific point, you can look at the second derivative of the function at that point. If the graph of the first derivative is flat, it means that the original function is neither concave up nor concave down at that point, possibly indicating a point of inflection or a horizontal tangent line. The first derivative can also be used to find the exact
  • #1
Econometricia
33
0
1. I need to find where the graph of f is CU or CD given the graph of f ' .



2. So far I have found that F inc. on (2,4) and (6,9). F has a local max at 4. F has a local min at 2 and 6.
 
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  • #2
When f' is increasing, i.e. when f''>0, f is concave up, and conversely.
 
  • #3
you have to find f '' then you can judge concavity
if f'' is bigger than 0 then it's concave up if f'' is smaller than 0 than it is concave down
 
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1. How do you determine if a function is concave up or concave down using its first derivative graph?

In order to determine if a function is concave up or concave down using its first derivative graph, you need to look at the direction of the graph. If the graph of the first derivative is increasing, the original function is concave up. If the graph of the first derivative is decreasing, the original function is concave down.

2. Can a function be both concave up and concave down?

No, a function cannot be both concave up and concave down at the same time. A function can only have one type of concavity at a particular point.

3. How can you tell if a function is concave up or concave down at a specific point?

You can tell if a function is concave up or concave down at a specific point by looking at the second derivative of the function at that point. If the second derivative is positive, the function is concave up. If the second derivative is negative, the function is concave down.

4. What does it mean if the graph of the first derivative is flat?

If the graph of the first derivative is flat, it means that the original function is neither concave up nor concave down at that point. This could indicate a point of inflection or a horizontal tangent line.

5. Can you use the first derivative to find the exact point of inflection?

Yes, you can use the first derivative to find the exact point of inflection. The point of inflection occurs when the concavity of the function changes, which is represented by a sign change in the first derivative. By setting the first derivative equal to zero and solving for x, you can find the x-coordinate of the point of inflection.

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