Question

If $f^{\prime }\left(c\right)=f^{\prime }\left(c\right)=.......=f^{n-1}\left(c\right)=0$, $f^n\left(c\right)\ne 0$ and $n$ is even, then-

• A. $f\left(c\right)$ is neither a maximum nor a minimum values of $f\left(x\right)$
• B. $f\left(c\right)$ is a maximum value of $f\left(x\right)$ if $f^n\left(c\right)<0$
• C. $f\left(c\right)$ is minimum value of $f\left(x\right)$ if $f^n\left(c\right)<0$
• D. $f\left(c\right)$ is a maximum value of $f\left(x\right)$ is $f^n\left(c\right)>0$

Answer 1

The correct option is B $f\left(c\right)$ is a maximum value of $f\left(x\right)$ if $f^n\left(c\right)<0$

Considering nth derivative test.
Now it is given that n is even.
Hence
$f\left(c\right)$ is relative maximum if $f^n\left(c\right)<0$ .