I was messing around with a function but cannot seem to get the intervals where it assumes negative values right.
Let $f(x)=3x^\frac{2}{3}-x$. Find the relative extrema of $f(x)$ in $[-1;27]$.
What I did was try to get the intervals where its first derivative assumes positive and negative values ($f'(x)>0$ and $f'(x)<0$ respectively), roots ($f'(x)=0$) and other critical points ($f'(x)$ Does Not Exist).
I got $x<8 \Rightarrow f'(x)>0$ and $ x > 8 \Rightarrow f'(x)<0$. $x=8$ is obvs. a maximum and $x=0 \Rightarrow \nexists f'(0)$.
I'm having trouble showing that $f'(x)$ assumes negative values in the interval $[-1; 0)$. Any way I can get that out of an equation?