Question

If $f\left(x\right)$ is a continuous and differentiable function such that the equation ${\int }_0^{x}sinf(x-t)dt=f\left(x\right)-sinx$ holds for any real number $x$, then

• A. $f^{\prime }\left(2\right)=3$
• B. $f^{\prime }\left(3\right)=1$
• C. $f\left(2\right)=2$
• D. $f^{\prime }\left(1\right)=0$

Answer 1

Answer: (B), (C)

$f\left(2\right)=2$
$f^{\prime }\left(3\right)=1$

${\int }_0^{x}sintf(x-t)dt=f\left(x\right)-sinx$ .......$\left(i\right)$
Substitute $x=0\Rightarrow f\left(0\right)=0$
$\Rightarrow {\int }_0^{x}sin(x-t)f\left(t\right)dt=f\left(x\right)-sinx$ .......$\left(ii\right)$
on differentiating
${\int }_0^{x}cos(x-t)f\left(t\right)dt=f^{\prime }\left(x\right)-cosx$ .......$\left(iii\right)$
Substitute $x=0$ in equation $\left(iii\right)$
$\Rightarrow f^{\prime }\left(x\right)=1$
$\Rightarrow f\left(x\right)=x+C$
$\Rightarrow f\left(x\right)=x$ $\because f\left(0\right)=0$
$f\left(2\right)=2$
$f^{\prime }\left(3\right)=1$