Hello. Please see attached.
1. How many points of inflection are on the blue function shown? Is it 1, 2, or 3? I read somewhere that where f''(x)=0 is a point of inflection (not necessarily a stationary point of inflection).
2. Regardless, what is the formal definition of a point of inflection?
Can anyone shed any light onto this?
Note: I've edited this post as before it didn't make sense and I didn't want to potentially mislead anybody.
Considering you are using second derivative test I'm assuming you have knowledge about concaves.
Point of inflection is defined as a point at which the concavity of the graph changes, the point at which a function transfers from concave up to concave down (or vice versa).
However, please note \(f''(x)=0\) does not always result to point of inflection it can be local max, local, min or inflection. You need to further consider it's domain and if satisfies the above definition of an inflection point. For example, \(f(x)=x^4\) has second derivative of \(f''(x)=12x^2\) and from it's first derivative, we can see it's stationary point is located at \((0,0)\), so we sub that in the second derivative to see it's nature and we get \(f''(x)=12x^2=12(0)^2=0\). Some students may just assume straight away it has a stationary point of inflection at \(x=0\), however considering domains (\(f''(x)>0\) for all \(x\)) you can see that it's second derivative is strictly positive meaning the whole graph is concave up and therefore can never have a point at which changes from concave up to concave down (or vice versa).
Note: Second derivative test is
not in the study design, you are
not required to know about concavity in methods, you are welcomed to use the textbook method that determines the nature of stationary point by examining the first derivative around that point. Some internal assessments may want you to use the textbook method.
Finding the nature of a stationary point by using first derivative is like so:
If we have a stationary point at \(x=0\) and we have \(f'(x)<0\) for when \(x<c\) and \(f'(x)>0\) for when \(x>c\) that will give a local min. And if we have \(f'(x)>0\) for when \(x<c\) and \(f'(x)<0\) for when \(x>c\) will give a local max. And if we have \(f'(x)>0\) for when \(x<c\) and \(f'(x)>0\) for when \(x>c\) we have a stationary point of inflection.
Refer to 2010 MCQ 17 if you want a question where you
need to use the textbook (first derivative) method.
If it is more convenient for you, you can draw up a increase/decreasing diagram. For a local min the diagram will look like this \_/
For a local max the diagram will look like this /-\ and for an inflection it should look like this \_\ or /-/ (kind of like a linear cubic wave). Where _ represents 0 gradient, / represents positive gradient and \ represents negative gradient. (like a linear function would).