Question

Let $f ( x )$ is a twice differentiable function, such that $f (1-2 x )= f (1+2 x ) \forall x \in R$, then minimum number of roots of equation $\left(f^{\prime \prime}(x)\right)^2+f^{\prime}(x) \cdot f^{\prime \prime \prime}(x)=0$ in $x \in(-5,10)$ is (given that $f(2)=f(5)=f(10)$ ) is

• A. 2
• B. 4
• C. 6
• D. 8

Answer 1

Answer: (C)

$ f (2)= f (0)$
and $f(5)=f(-3)$
and $f (10)= f (-4)$
$\therefore f ^{\prime}( x )=0 \rightarrow \min .4$ times
and $f ^{\prime \prime}( x )=0 \rightarrow \min .3$ times
$\therefore f ^{\prime}( x ) \cdot f ^{\prime \prime}( x )=0 \rightarrow \min .7$ times.