Question

Let $f\left(x\right)$ be a function such that $f^{\prime }\left(a\right)\ne 0$.Then at $x=a,f\left(x\right)$

• A. cannot have a maximum
• B. cannot have a minimum
• C. must have neither a maximum nor a minimum
• D. none of these

Answer 1

The correct option is D none of these

Consider a function $y=f\left(x\right)$.

A function achieves a maximum value at a, if

$f^{\prime }\left(a\right)=0$ and $f^{\prime \prime }\left(a\right)<0$

Similarly a function achieves a minimum value at a if

$f^{\prime }\left(a\right)=0$ and $f^{\prime \prime }\left(a\right)>0$

If $f^{\prime }\left(a\right)\ne 0$, we cannot definitely say anything about maxima or minima of a function.