# Second derivative test

1. Jun 6, 2015

The second derivative test can only be applied if $f''$ is continuous in a region around $c$.
But according to this link: http://calculus.subwiki.org/wiki/Second_derivative_test#Requirement_of_twice_differentiability
$f''$ need not be defined in a region around $c$.
I'm confused as to what is required for the second derivative test.
$f''$ is allowed to have a discontinuity at $c$, but not around $c$? If $f''$ is continuous around $c$, then $\lim_{x→a} f''(x)$ must equal $f''(a)$ where $a$ is in some region around $c$, but according to the second link, $f''(a)$ doesn't have to be defined.

2. Jun 6, 2015

### wabbit

There is no need for $f''$ to be defined at points other than c. If $f'(c)=0$ and $f''(c)>0$, using the definition $f''(c)=\lim_{h\rightarrow 0}\frac{f'(c+h)-f'(c)}{h}$ this implies that within some neighborhood of $c,f'$ is positive to the right of of $c$ and negative to its left, so f must have a minimum at $c$ .

3. Jun 6, 2015

http://tutorial.math.lamar.edu/Classes/CalcI/DerivativeAppsProofs.aspx#Extras_DerAppPf_SDT

The author assumes that $f''$ is continuous around $c$ to prove the second derivative test.

4. Jun 6, 2015

### MrAnchovy

Only that $f''$ is defined at $c$. I don't see where in the first link it suggests a requirement for continuity: if it does it is wrong.

5. Jun 6, 2015

### wabbit

I just gave you a proof that does not require that assumption. Did you read it ? If you don't understand it I can help you with fleshing out the details.

Whatever the author assumes is irrelevant to the question you asked about which assumption is necessary.

Last edited: Jun 6, 2015