Proving an Inflection Point

[student1081]

Hey guys, so I got a question...
If a question tells me that I need to prove that the point is an inflection point, after telling me to find the second derivative and solve it for 0, do I need to draw up the table to prove the inflection point?
Many thanks for any responses

[RuiAce]

Yes you do. The second derivative test for inflection points isn't as 'convenient' as the first derivative test for stationary points. You must always draw up the table checking a bit to both sides to test for an explicit change in concavity.

(The reason is because the implication is one-sided. It is true that if \(x_0\) is an inflection point, then \( f^{\prime\prime}(x_0) = 0\). But the converse fails. That is, if \(f^{\prime\prime}(x_0) = 0\), then there may or may not be an inflection point at \(x_0\). The most famous example of this is \(f(x) = x^4\), at \(x=0\).)

[student1081]

Many thanks for the quick response!