Question

Let $f(x)$ and $g(x)$ be twice differentiable function in R and satisfying the equation $f^{\prime \prime}(x)=g^{\prime \prime}(x)$ such that $f^{\prime}(1)=2 g^{\prime}(1)=4$ and $f(2)=3 g(2)=9$, then
[where $\operatorname{sgn} x$ dentoes the signum function of $x$.]

• A. $f(x)+g(x)=3 x+5$
• B. if $|f(x)-g(x)|<1$ then sgn $x=-1$
• C. $f(3)-g(3)=8$
• D. area bounded between the curve $y=f(x)$ and $y=g(x)$ in $[0,2]$ is 8 .

Answer 1

Answer: (B), (C), (D)

Given$ f^{\prime \prime}(x)=g^{\prime \prime}(x) $
$f ^{\prime}( x )= g ^{\prime}( x )+\lambda$....(i)
$\text { and } f(x)=g(x)+\lambda x+\mu$....(ii)
$\text { According to question } $
$f^{\prime}(1)=4, g^{\prime}(1)=2$....(iii)
$\text { and } f(2)=9 \text { and } g(2)=3 $....(iv)
$\because \text { From (i), (ii), (iii) and (iv) } $
$\lambda=4=2 $
$\therefore f ( x )= g ( x )+2 x +2 $