Problem

To determine if $f(x)$ has a local extremum at $x = 1$, you need to check whether the derivative equals zero there and whether the derivative actually changes sign around that point. Statement II tells you that $f'(1) = 0$, which is necessary but not sufficient—a critical point is only a local extremum if the function changes from increasing to decreasing (or vice versa). The key is to examine the sign of $f'(x)$ on either side of $x = 1$ using the product rule, or to check the second derivative test, to confirm whether this critical point is truly an extremum.

To determine if $f(x)$ has a local extremum at $x = 1$, you need to check whether the derivative equals zero there and whether the derivative actually changes sign around that point. Statement II tells you that $f'(1) = 0$, which is necessary but not sufficient—a critical point is only a local extremum if the function changes from increasing to decreasing (or vice versa). The key is to examine the sign of $f'(x)$ on either side of $x = 1$ using the product rule, or to check the second derivative test, to confirm whether this critical point is truly an extremum.