A curve's inflection point is the point at which the curve's concavity changes. For a function its concavity can be measured by its second order derivative When which means that the function's rate of change is decreasing, the function is concave down. In contrast, when the function's rate of change is increasing, i.e. the function is concave up.
In typical problems, we find a function's inflection point by using provided that and are both differentiable at that point and checking the sign of around that point. Be careful not to forget that does not necessarily mean that the point is an inflection point since the sign of might not change before and after that point.
In the figure above, the red zone depicts the area where the function is concave down and the blue zone indicates concave up.
What are the inflection points of the curve
Checking the signs of around and we get the table below:
This table tells us that is concave up for concave down for and concave up for Hence, the two inflection points of the curve are and or equivalently,
What is the slope of the tangent of the curve at its inflection point?
The values of and are both at Checking the signs of and around we get the table below:
The swithcing signs of in the table tells us that is concave down for and concave up for implying that the point is the inflection point of the graph Since the table also tells us that the slope of the tangent of at its inflection point is
How many inflection points does have in the interval
Let Then, differentiating twice gives
Since it is true that Thus, is either zero or positive, so the sign of does not change. Therefore, has no inflection points in the interval
If how many inflection points does the function have?
The second order derivative of is
Thus, at and Checking the signs of around and we get the table below:
Since the sign of does not change before and after the function only has an inflection point at Therefore the answer is 1.