Question

Let $f:[a,b]\to R$ be a function such that $c\in (a,b),f^{\prime }\left(c\right)=f"\left(c\right)=f""\left(c\right)=f^{iv}\left(c\right)=f^v\left(c\right)=0$, then:

• A. $f$ has local extremum at $x=c$
• B. $f$ has neither local maximum nor local minimum at $x=c$
• C. $f$ is necessarily a constant function
• D. it is difficult to say whether ($f$ has local extremum at $x=c$) or ($f$ has neither local maximum nor local minimum at $x=c$)

Answer 1

Answer: (A)

$f$ has local extremum at $x=c$

$\because f^i\left(C\right)=f^{ii}\left(C\right)=fiiii\left(C\right)=f^{iv}\left(C\right)=f^v\left(C\right)=0$

so, following nth derivative test, we will get $f^{\left(n\right)}\left(C\right)$ where it is not equals to zero.

so, f has local extremum at x=c.