Differential Calculus
Differential Calculus
star_batch_jee_advanced_2025
Grade 12
Question:
Let $f(x) = \cos x$ and $H(x) = \begin{cases} \min(f(t): 0 \leq t 0 \\ 1, & \text{for } x = 0, \text{ where } [x] \text{ is greatest integer function and } \{x\} \text{ is the fractional part} \\ \sqrt{|x|}\cot\{x\}, & \text{for } x 0$, let $l = \lim_{x \to \infty} \frac{e^{ax} - e^{ax}}{\cot x - \cos x}$ and $m = \lim_{x \to \infty}\left(\sqrt{x^2 + ax} - \sqrt{x^2 - ax}\right)$, then:
l > m, for all a > 0
l > m, when a >= 1
l > m, for all a > e^(-a)
e^l + m = 0
Step-by-Step Solution
Key Concept: A piecewise function is differentiable at a junction point when the one-sided derivatives match.
Given $H(x) = \begin{cases} \cos x, & 0 \leq x < \frac{\pi}{2} \\ \frac{\pi}{2} - x, & \frac{\pi}{2} \leq x \leq 3 \end{cases}$, we find $H'\left(\frac{\pi}{2}^-\right) = -\sin x = -1$ and $H'\left(\frac{\pi}{2}^+\right) = -1$. Since both one-sided derivatives equal $-1$, $H(x)$ is continuous and differentiable on $[0,3]$. The maximum value is $H(0) = 1$.
Correct Answer: 1,4