Question

$f\left(x\right),g\left(x\right)$ are two differentiable function on $[0,2]$ such that $f^{\prime \prime }\left(x\right)-g^{\prime \prime }\left(x\right)=0$ and $f^{\prime }\left(1\right)=4=2g^{\prime }\left(1\right)$ and $f\left(2\right)=3g\left(2\right)=9$ then $[f\left(x\right)-g\left(x\right)]$ at $x=\frac{3}{2}$ is

• A. $0$
• B. $2$
• C. $10$
• D. $5$

Answer 1

The correct option is D $5$

$f^{\prime \prime }\left(x\right)-g^{\prime \prime }\left(x\right)=0$
Integrating we get, $f^{\prime }\left(x\right)-g^{\prime }\left(x\right)=k$
$\Rightarrow f^{\prime }\left(1\right)-g^{\prime }\left(1\right)=k$ $\therefore k=2$
so $f^{\prime }\left(x\right)-g^{\prime }\left(x\right)=2$
Again integrating $f\left(x\right)-g\left(x\right)=2x+k_1$
$\Rightarrow f\left(2\right)-g\left(2\right)=4+k_1\therefore k_1=2$
Hence $f\left(x\right)-g\left(x\right)=2x+2$
$\therefore {[f\left(x\right)-g\left(x\right)]}_{x=\frac{3}{2}}=\frac{2\times 3}{2}+2=5$