Question

Consider a real valued continuous function $f$ such that $f(x)=\sin x+\underset{-\pi /2}{\overset{\pi /2}{\int }}(\sin x+tf(t))dt$. If and $m$ are maximum and minimum value of the function $f$, then

• A. $\frac{M}{m}=3$
• B. $M-m=2\pi +1$
• C. $M+m=4(\pi +1)$
• D. $Mm=2\left({\pi }^2+1\right)$

Answer 1

Answer: (A), (C)

$f(x)=\sin x+\pi \sin x+\underset{-\pi /2}{\overset{\pi /2}{\int }}tf(t)dt=(\pi +1)\sin x+A$
$A=\underset{-\pi /2}{\overset{\pi /2}{\int }}((\pi +1)\sin t+A)tdt$
$A=2(\pi +1)\underset{0}{\overset{\pi /2}{\int }}t\sin tdt={2(\pi +1)(-\cos t\cdot t+\sin t)|}_0^{\pi /2}$
$=2(\pi +1)(1)=(\pi +1)\cdot 2$
$f(x)=(\pi +1)(2+\sin x)$
$M=3(\pi +1),m=(\pi +1)$