Differentiation and its Application: Let f:[0,infty)to[0,3] be a function defined by… | JEE Main 2021

JEE Main 2021 (Online) 27th July Evening ShiftQuestion 1

Let $f:[0,\infty)\to[0,3]$ be a function defined by $f(x)=\begin{cases}\max\{\sin t:0\le t\le x\}, & 0\le x\le\pi\\ 2+\cos x, & x>\pi\end{cases}$. Then which of the following is true ?

Differentiation and its Application

$f$ is not continuous exactly at two points in $(0,\infty)$$f$ is continuous everywhere but not differentiable exactly at two points in $(0,\infty)$$f$ is continuous everywhere but not differentiable exactly at one point in $(0,\infty)$$f$ is differentiable everywhere in $(0,\infty)$✓ Correct answer

Solution

Concept. Write the running maximum of $\sin t$ piece by piece, then check slopes at $\frac\pi2$ and $\pi$. Formulas used. (1) $\sin$ increases on $\left[0,\frac\pi2\right]$ up to $1$ and then fallsStep 1 - Pieces. By (1): $f=\sin x$ on $\left[0,\frac\pi2\right]$, $f=1$ on $\left[\frac\pi2,\pi\right]$, $f=2+\cos x$ for $x>\pi$. Step 2 - At $\frac\pi2$. Value $1$ both sides; slopes $\cos\frac\pi2=0$ and $0$. Differentiable. Step 3 - At $\pi$. $2+\cos\pi=1$, so continuous; slopes $0$ and $-\sin\pi=0$. Differentiable. Step 4 - Conclude. $f$ is differentiable everywhere in $(0,\infty)$. Answer: (D)

Differentiation and its Application - JEE Mathematics question (JEE Main 2021 (Online) 27th July Evening Shift)

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