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

$(a)\;0\qquad(b)\;2\qquad(c)\;10\qquad(d)\;5$

please add the required comma in the question after 4,f(2) it ruined my 30 minutes

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A)
$f''(x)-g''(x)=0$
Integrating $f'(x)-g'(x)=c$
$f'(1)-g'(1)=c$
$\Rightarrow 4-2=c$
$c=2$
$f'(x)-g'(x)=2$
Integrating $f(x)-g(x)=2x+c_1$
$\Rightarrow f(2)-g(2)=4+c_1$
$\Rightarrow 9-3=4+c_1$
$c_1=2$
$f(x)-g(x)=2x+2$
At $x=\large\frac{3}{2}$
$f(x)-g(x)=3+2=5$
Hence (d) is the correct answer.