JEE Advanced 2026 · Paper 1, Mathematics question 1

Derivatives.
$$f'(x)=\frac{\ln x+2}{2\sqrt x}-1,\qquad f''(x)=-\frac{\ln x}{4x^{3/2}}.$$

• $f''>0$ on $(0,1)$, so $f'$ is increasing there. (A) false.
• $f''<0$ on $(1,\infty)$, so $f'$ has its maximum at $x=1$, where $f'(1)=1-1=0$.

So $f'(x)\le0$ everywhere, with equality only at $x=1$. Then $f$ is strictly decreasing and $f'$ does not change sign at $x=1$, so $f$ has no local maximum or minimum. (D) true, and (B), (C) are false.