Question

If $f(0)=f(1)=f(2)=0$ & function $f(x)$ is twice differentiable in $(0,2)$ and continuous in $[0,2]$. Then which of the following is/are definitely true -

• A. $f^{\prime \prime }(c)=0;\forall c\in (0,2)$
• B. $f^{\prime }(c)=0$; for atleast two $c\in (0,2)$
• C. $f^{\prime }(c)=0$; for exactly one $c\subset (0,2)$
• D. $f^{\prime \prime }(c)=0$; for atleast one $c\subset (0,2)$

Answer 1

Answer: (B), (D)

$\because f(0)=f(1)\&{}^{\prime }{}^{\prime }$ is continuous in $[0,1]\&$ derivable in $(0,1)$
$\therefore f^{\prime }\left(c_1\right)=0$ for atleast one $c_1\in (0,1)$
similarly,
$\because f(1)=f(2)$
$\therefore f^{\prime }\left(c_2\right)=0$ for atleast one $c_2\in (1,2)$
$\Rightarrow f^{\prime }\left(c_1\right)=f^{\prime }\left(c_2\right)$
$\Rightarrow f^{\prime \prime }(c)=0$ for atleast one $c\in \left(c_1,c_2\right)$