JEE Advanced 2017 · Paper 1, Mathematics question 4

Here $0<f<1$ everywhere.

• (A) $g(x)=x^9-f(x)$ has $g(0)=-f(0)<0$ and $g(1)=1-f(1)>0$. By the IVT it has a zero in $(0,1)$. True.

• (B) $h(x)=x-\displaystyle\int_0^{\frac\pi2-x}f(t)\cos t\,dt$:

  • $h(0)=-\displaystyle\int_0^{\pi/2}f\cos t\,dt<0$;
  • $h(1)>1-\displaystyle\int_0^{\frac\pi2-1}\cos t\,dt=1-\cos1>0$.

  So it has a zero. True.

• (C) $\displaystyle\int_0^xf\sin t\,dt<\int_0^1\sin t\,dt<1<e^x$, so the function stays positive. False.

• (D) It is a sum of positive terms. False.

(A), (B)