JEE Advanced 2025 · Paper 2, Mathematics question 8

Rewrite f.
$$f(x)=1+\frac{4}{2+h(x)},\qquad h(x)=\frac{\sin x}{x}.$$
So $f$ rises exactly where $h$ falls: maxima of $f$ are minima of $h$, and minima of $f$ are maxima of $h$.

At x = 0. $h<1=h(0)$ near 0, so $f>\frac73=f(0)$. Hence $x=0$ is a local minimum: (B) is true and (A) is false.

Other extrema. For $x>0$, the extrema of $h$ occur where $\tan x=x$, at one root in each interval $(n\pi,\,n\pi+\frac\pi2)$:

• $n$ odd ($\sin x<0$): $h$ has a minimum, so $f$ has a maximum.
• $n$ even: $h$ has a maximum, so $f$ has a minimum.

(C) On $[\pi,6\pi]$ the maxima of $f$ come from $n=1,3,5$: 3 points. ✓

(D) On $[2\pi,4\pi]$ the only minimum of $f$ comes from $n=2$ (the $n=4$ root lies beyond $4\pi$): 1 point. ✓