- Home
- Standard 12
- Mathematics
5. Continuity and Differentiation
normal
If $f$ is a differentiable function such that $f(2x + 1) = f(1 -2x)$ $\forall \,\,x \in R$ then minimum number of roots of the equation $f'(x) = 0$ in $x \in \left( { - 5,10} \right)$ ,given that $f(2) = f(5) = f(10)$ , is
A
$2$
B
$3$
C
$4$
D
$5$
Solution
$f(2 \mathrm{x}+1)=f(1-2 \mathrm{x})$
$\Rightarrow f(1+x)=f(1-x)$ symmetric about the line $x=1$
$\therefore \quad f(2)=f(0) ; f(5)=f(-3)$ and $f(10)=f(-8)$
$\Rightarrow f(-3)=f(0)=f(2)=f(5)=f(10)$
$\therefore $ according to $RMVT$ $f^{\prime}(\mathrm{x})=0$ have at least
$4$ roots in $\mathrm{x} \in(-5,10).$
Standard 12
Mathematics
