Question

If $f^{\prime }\left(c\right)<0$ and $f^{\prime }\left(c\right)>0$, then at $x=c$, $f\left(x\right)$ is-

• A. maximum
• B. minimum
• C. neither maximum nor minimum
• D. not possible

Answer 1

The correct option is D not possible

$f\left(c\right)$ is the maximum value of $y=f\left(x\right)$ if
$f^{\prime }\left(c\right)=0$ and
$f^{\prime \prime }\left(c\right)<0$.
And
$f\left(c\right)$ is the minimum value of $y=f\left(x\right)$ if
$f^{\prime }\left(c\right)=0$ and
$f^{\prime \prime }\left(c\right)>0$
Now
$f^{\prime }\left(c\right)$ denotes the slope of $y=f\left(x\right)$ at $x=c$.
That is
$f^{\prime }\left(c\right)$ denotes $\tan \theta$ which denotes direction of the $y=f\left(x\right)$ at $x=c$ and hence cannot be negative and positive at the same time.
Hence the above situation is not possible.