Question

If $f^{\prime \prime }\left(x\right)<0$ for all $x\in (a,b)$, then $f^{\prime }\left(x\right)=0$

• A. exactly once in $(a,b)$
• B. at most once in $(a,b)$
• C. at least once in $(a,b)$
• D. none of these

Answer 1

The correct option is B at most once in $(a,b)$

Let $f^{\prime }\left(x\right)=0$ be at two distinct points $x_1,x_2$ in (a,b).
$\Rightarrow f^{\prime }\left(x_1\right)=0$
$\Rightarrow f^{\prime }\left(x_2\right)=0$
Then, by Rolle's theorem, there exists c$\in$ (a,b) such that
$\Rightarrow f^{\prime \prime }\left(c\right)=0$
which contradicts the given statement f''(x)<0
Hence, our assumption is wrong.
Hence, there can be at most one point in (a,b) such that $f^{\prime }\left(x\right)=0$