# Given the first and second derivatives, determine whether it is an local extrmum

If $f'(x)=0$ and $f''(x)\neq0$, does it mean that function $f$ has a local extremum in $x$?
If $f'(x)=0$ and $f''(x)=0$, does it mean that function $f$ has no local extremum in $x$?

• Why don't you start with some examples like $f(x) = x^2 ; x^3$ which may gives you some insight into a more formal solution. – Ram Feb 10 '13 at 12:39

If $f'(x)=0$ and $f''(x)\neq0$, then there is a local extremum in $x$. When $f''(x)=0$, there is a horizontal slope and an inflection point, and not necessarily an extremum. Think of $f(x)=x^3$. As @DavidMitra points out in the comment below, this reasoning applies only to odd powers. For even powers, there is a local extremum.
• What about $f(x)=x^4$? This does not have an inflection point at $x=0$, it does have an extremum at $x=0$, yet $f'(x)=f''(x)=0$. – David Mitra Feb 10 '13 at 13:35