Free JEE Main Maths PYQ Continuity for Composition and Function MCQs with step-by-step solutions (1 questions). Part of Limit, Continuity and Differentiability. Practise online on Prepizo — no login needed.
Q1 — Continuity for Composition and Function · hard
For the function $f(x) = x\cos\dfrac{1}{x}$, $x \ge 1$,
A. for at least one $x$ in the interval $[1, \infty)$, $f(x+2) - f(x) = 2$
B. $\displaystyle\lim_{x\to\infty} f'(x) = 1$ ✓ Correct
C. for all $x$ in the interval $[1, \infty)$, $f(x+2) - f(x) < 2$ ✓ Correct
D. $f'(x)$ is strictly decreasing in the interval $[1, \infty)$ ✓ Correct
Solution: $f'(x) = \cos\frac{1}{x} + \frac{1}{x}\sin\frac{1}{x} \to \cos 0 + 0 = 1$ as $x\to\infty$, so (b) holds. Since $f'(x) < 1$ throughout $[1,\infty)$ and $f'$ can be shown strictly decreasing there, the Mean Value Theorem gives $f(x+2)-f(x) < 2$ for every $x \ge 1$, so (c) and (d) hold, while equality in (a) is never attained.