Video: Finding an Inflection Point From the Graph of the Second Derivative

The graph shows 𝑓″(π‘₯), the second derivative of 𝑓(π‘₯), with horizontal tangents at π‘₯ = βˆ’βˆš3 and π‘₯ = √3. For what values of π‘₯ does 𝑓(π‘₯) have an inflection point?

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Video Transcript

The graph shows 𝑓 double prime of π‘₯, the second derivative of 𝑓 of π‘₯, with horizontal tangents at π‘₯ equals negative root three and π‘₯ equals root three. For what values of π‘₯ does 𝑓 of π‘₯ have an inflection point?

Remember, an inflection point is a point in the graph at which the concavity changes. In other words, the graph goes from being concave down to concave up or vice versa. Now, if the second derivative 𝑓 double prime of π‘₯ is less than zero over some interval, then the graph is concave down over that same interval. Conversely, if the second derivative 𝑓 double prime of π‘₯ is greater than zero, then the graph is concave up.

Now, if the second derivative 𝑓 double prime of π‘₯ is equal to zero, that can indicate that we have an inflection point but not necessarily. So, instead, we look for where the concavity changes, in other words, where the second derivative goes from being positive to negative or vice versa.

Well, we can see that the first point at which this occurs is when π‘₯ is equal to negative three. As we approach negative three from the left, the second derivative is less than zero. Whereas as we approach it from the right, the second derivative is greater than zero. Similarly, as we approach π‘₯ equals zero from the left, the second derivative is greater than zero. Whereas as we approach it from the right, the second derivative is less than zero.

There’s one other place where this happens. And that’s when π‘₯ is equal to three. As we approach three from the left, the second derivative 𝑓 double prime is less than zero. Whereas as we approach it from the right, the second derivative is greater than zero. We can, therefore, say that the concavity of the graph 𝑓 of π‘₯ changes at π‘₯ equals negative three, zero, and three. And this, in turn, means that these are each inflection points. So, the answer here is negative three, zero, and three.

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