Question

Let $f$ be any continuously differentiable function on $[a,b]$ and twice differentiable on $(a,b)$ such that $f\left(a\right)=f^{\prime }\left(a\right)=0$ and $f\left(b\right)=0$. Then

• A. $f^{\prime \prime }\left(a\right)=0$
  
• B. $f^{\prime }\left(x\right)=0$ for some $x\in (a,b)$
  
• C. $f^{\prime \prime }\left(x\right)=0$ for some $x\in (a,b)$
• D. $f^{\prime \prime \prime }\left(x\right)=0$ for some $x\in (a,b)$

Answer 1

Answer: (B), (C)

The correct options are
B $f^{\prime }\left(x\right)=0$ for some $x\in (a,b)$

C $f^{\prime \prime }\left(x\right)=0$ for some $x\in (a,b)$

Applying Rolle's theorem to $f$ on the interval $[a,b]$, we get
$f^{\prime }\left(c\right)=0$ for some $c\in (a,b)$

Again $f^{\prime }\left(a\right)=0=f^{\prime }\left(c\right)$ for some $x\in (a,c)\subset (a,b)$
Therefore, Rolle's theorem is applicable to $f^{\prime }$ on the interval $(a,c)$
Hence, $f^{\prime \prime }\left(c_1\right)=0$ for some $c_1\in (a,c)$