Question

Consider a real valued continuous function $f$ such that $f(x)=\sin x+\int\limits_{-\pi / 2}^{\pi / 2}(\sin x+t f(t)) d t$. 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 +\int\limits_{-\pi / 2}^{\pi / 2} tf ( t ) dt =(\pi+1) \sin x + A$
$A=\int\limits_{-\pi / 2}^{\pi / 2}((\pi+1) \sin t+A) t d t$
$A =2(\pi+1) \int\limits_0^{\pi / 2} t \sin t d t =\left.2(\pi+1)(-\cos t \cdot t +\sin t )\right|_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) $