On a homework problem, I got the wrong answer and figured out what to put in for it to be marked correct (online homework), but I am unsure why it is right.
The problem is to find the critical numbers for $f(x)=x^{-5}\log x$. Critical numbers occur where the derivative is 0 or undefined, so the first step is to find the derivative.
Product rule: $-5(x^{-6})\ln(x) + (x^{-5})(1/x)$
Simplify: $-5(x^{-6})\ln(x) + (x^{-6})$
Factor out $(x^{-6})$: $(x^{-6})(-5\ln(x)+1)$
$x^{-6}=1/(x^6)$ which will be undefined at $0$, so that should be part of the list of critical numbers.
Now to find zeros of $-5\ln(x)+1$: $-5\ln(x)+1=0$, $1=5\ln(x)$, $1/5=\ln(x)$, $x=e^{1/5}$.
I put in the list $0,e^{1/5}$ and it was marked incorrect.
On a hunch I removed $0$ from the list. My answer was marked correct.
Isn't $f'(x)$ undefined at $x=0$, or am I missing something? I have been marked correct on other answers which listed points where the derivative is not defined, so I am sure my definition of critical numbers is correct.