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 $x$?

If $f'(x)=0$ and $f''(x)\
eq0$, 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.

For surfaces in two dimensions, places where the gradient vanishes, as well in places where either the second partial derivatives vanish or a matrix of second derivatives (known as a Hessian) is indefinite is called a saddle point.