Science:Math Exam Resources/Courses/MATH102/December 2016/Question A 07/Statement

Let $f(x)$ be a function that is continuous at $x=0$. Which of the following statements is correct? Recall that if there exists any function for which a statement is false then the statement is false.

(i) If $f'(0)$ exists and $f'(0)=0$ then $x=0$ is an extremum.

(ii) If $x=0$ is an extremum then $f'(0)$ exists and $f'(0)=0.$

(iii) If $f'(x)$ exists for all $x$ and $f'(0)=0$, $f'(1)>0$, and $f'(-1)<0$ then $x=0$ is a minimum.

(iv) If $f'(0)$, $f''(0)$ and $f'''(0)$ exist and $f'(0)=f''(0)=0\neq f'''(0)$ then $x=0$ is not an extremum.

(v) None of the above are true.