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 $3\le x\le 7$
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

Answer: 152

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


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