second derivative

[burt]

Can there be a point a on the graph f(x) such that a is an inflection point on the graph, but f"(a) does not equal zero?

 

[Dr.Peterson]

I presume you mean f"(a).

Yes, it is possible that f"(a) does not exist.

 

[burt]

can you give me an example?

 

[burt]

Also, is a graph concave up whenever it's second derivative is positive?

 

[Dr.Peterson]

A classic example is the piecewise function f(x) = x^2 for x >= 0, and -x^2 for x < 0.

What does your textbook say about points of inflection, and about concave up functions? Quote from it, and we can discuss that.

 

[pka]

Also, is a graph concave up whenever it's second derivative is positive?

Have you considered something like this?
\(\displaystyle f(x)=\begin{cases}-(x+1)^2+1 &: x\le 0 \\ (x-1)^2-1 &: x>0\end{cases}\)

Is \(\displaystyle x-0\) a point of inflection? Now as Prof. Peterson points out it may well depend upon the definitions in your textbook.