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

Answer 1

Answer: (B), (C)

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

Applying Rolle's Theorem of $f\left(x\right)$
$f\left(a\right)=f\left(b\right)=0$ so $f^{\prime }\left(x\right)=0$ for some $x=cϵ(a,b)$ again $f^{\prime }\left(a\right)=0=f^{\prime }\left(c\right)$ so for some $xϵ(a,c)$ i.e. $(a,b),f"\left(x\right)=0$.