Question

Let the function $f$ and $g$ be defined by

Let A denotes the sum of all the solutions of the equation $f( x )=0.6$ for $3 \leq x \leq 7$
B denotes the fundamental period of $g ( x )$.
C denotes the value of $g^{\prime}$ (6.75)
Compute the value of $[ A ] BC$ where [ ] denotes the greatest integer function.

Answer 1

Since $f(x+2)=f(x)$
$\Rightarrow f$ is periodic with period $=2 \Rightarrow $ period of $g =2 / 3$ now graph of $y=f(x)$


$ g ( x )=4 f (3 x )+1 $
$C : g ^{\prime}( x )=12 f ^{\prime}(3 x ) $
$g ^{\prime}(6.75)=12 \cdot f ^{\prime}(3 \times 6.75)=12 f ^{\prime}(20.25)$
since fhas a period ' 2 '
$\therefore f ^{\prime}( x )$ also has a period ' 2 '
$f ^{\prime}(20+0.25)= f ^{\prime}(0.25)= f ^{\prime}(1 / 4)$
$\therefore g ^{\prime}(6.75)=12 f ^{\prime}(1 / 4) $
$f ( x )=\sqrt{ x } \text { in }(0,1) $
$f ^{\prime}( x )=\frac{1}{2 \sqrt{ x }} \Rightarrow f ^{\prime}\left(\frac{1}{4}\right)=1 $
$\Rightarrow g ^{\prime}(6.75)=12 $
$\therefore [ A ] BC =19 \times \frac{2}{3} \times 12=19 \times 8=152 $