If a saddle point is on the boundary, is it a local extremum?

Question

Consider the function $f(x)=x^4-4x^3$, which has a saddle point when $x=0$. Now, define the same function on $[0,10]$. How to characterize $x=0$ now? Is it a local extremum (local maximum)? I am confused because $f'(0)=0$ but $f'(0+ \epsilon)<0$.

Answer 1

Yes, if you restrict your function as you've said then it's a local maximum, similar to if you restricted it to $[1,10]$, then $x=1$ would be a local maximum.