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Q.
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 -
Application of Derivatives
Solution:
$\because f(0)=f(1) \& { }^{\prime}{ }^{\prime}$ 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 $c \in\left( c _1, c _2\right)$
