JEE Main 2018 (Online) 15th April Morning ShiftQuestion 1
Let $S=\left\{(\lambda,\mu)\in\mathbb R\times\mathbb R: f(t)=\left(|\lambda|e^{t}-\mu\right)\cdot\sin(2|t|),\ t\in\mathbb R,\text{ is a differentiable function}\right\}$. Then $S$ is a subset of
Differentiation and its Application
Solution
Concept. $\sin(2|t|)$ has a corner at $0$; multiplying by $g(t)=|\lambda|e^t-\mu$ removes it only if $g(0)=0$. Formulas used. (1) If $g$ is smooth, $g(t)\sin(2|t|)$ has one-sided derivatives $\pm2g(0)$ at $t=0$Step 1 - At $t\ne0$. Both factors are differentiable, so only $t=0$ matters. Step 2 - At $t=0$. By (1), the right derivative is $2g(0)$ and the left is $-2g(0)$. They agree only if $g(0)=0$: $$|\lambda|-\mu=0\ \Longrightarrow\ \mu=|\lambda|\ge0.$$ Step 3 - Conclude. Every $(\lambda,\mu)\in S$ has $\mu\ge0$, so $S\subseteq\mathbb R\times[0,\infty)$. Answer: (A)