Let f: [0, ∞) → [0, 3] be a function defined by f(x) = \ array ll max\ t: 0 ≤ t ≤ π\ , x ∈ [0, π] 2 + x, x > π array ). Then which of the following is true ?

Correct answer: B. f is differentiable everywhere in (0, ∞) Graph of max \ t: 0 ≤ t ≤ x\ in x ∈ [0, π] the graph of y = 2 + x for x ∈ [π, ∞) So graph of f(x) =\ cases x, & 0 ≤ x ≤ (π)/(2) 1, & (π)/(2) ≤ x ≤ π 2 + x, & x > π cases ) f (x) is differentiable everywhere in (0, ∞)

JEE Main 202127 JulyEvening Shift

Exam: JEE Main
Year: 2021
Date: 27 July
Shift: Evening Shift
Subject: Mathematics
Chapter: Continuity and Differentiability

JEE Main 2021 · 27 July · Evening Shift · Mathematics · Continuity and Differentiability

Let f: [0, ∞) → [0, 3] be a function defined by f(x) = \ array ll...

Let

f: \left[0, \infty\right) \to \left[0, 3\right] be a function defined by

f\left(x\right) = \left\{

\begin{array}{ll}max\left\{\sin t: 0 \le t \le \pi\right\}, x \in \left[0, \pi\right] \\ 2 + \cos x, x > \pi\end{array}

\right). Then which of the following is true ? $Let f: [0, ∞) → [0, 3] be a function defined by f(x) = \ array ll max\ t: 0 ≤ t ≤ π\ , x ∈ [0, π] 2 + x, x > π array ).$ $Let f: [0, ∞) → [0, 3] be a function defined by f(x) = \ array ll max\ t: 0 ≤ t ≤ π\ , x ∈ [0, π] 2 + x, x > π array ).$ $Let f: [0, ∞) → [0, 3] be a function defined by f(x) = \ array ll max\ t: 0 ≤ t ≤ π\ , x ∈ [0, π] 2 + x, x > π array ).$ $Let f: [0, ∞) → [0, 3] be a function defined by f(x) = \ array ll max\ t: 0 ≤ t ≤ π\ , x ∈ [0, π] 2 + x, x > π array ).$ $Let f: [0, ∞) → [0, 3] be a function defined by f(x) = \ array ll max\ t: 0 ≤ t ≤ π\ , x ∈ [0, π] 2 + x, x > π array ).$ $Let f: [0, ∞) → [0, 3] be a function defined by f(x) = \ array ll max\ t: 0 ≤ t ≤ π\ , x ∈ [0, π] 2 + x, x > π array ).$

Options

A $f$ is continuous everywhere but not differentiable exactly at one point in $\left(0, \infty\right)$
B $f$ is differentiable everywhere in $\left(0, \infty\right)$
C $f$ is not continuous exactly at two points in $\left(0, \infty\right)$
D $f$ is continuous everywhere but not differentiable exactly at two points in $\left(0, \infty\right)$

Correct answer

B. $f$ is differentiable everywhere in $\left(0, \infty\right)$

Step-by-step solution

Graph of $max \left\{\sin t: 0 ≤ t ≤ x\right\}$ in $x \in \left[0, \pi\right]<img alt="" src="/api/proxy-image?url=https%3A%2F%2Ffirebasestorage.googleapis.com%2Fv0%2Fb%2Fquantrexacademy-app.firebasestorage.app%2Fo%2Fquestions%252Ffigs%252Fpyq%252Fjee_main%252Fabb4563a-c619-42f1-ba79-d6cacc1c892a-05545eb3-cdc3-4eae-b4e0-1ca692972787-image.png%3Falt%3Dmedia&clean=1&v=wm2" data-qx-orig-src="https://firebasestorage.googleapis.com/v0/b/quantrexacademy-app.firebasestorage.app/o/questions%2Ffigs%2Fpyq%2Fjee_main%2Fabb4563a-c619-42f1-ba79-d6cacc1c892a-05545eb3-cdc3-4eae-b4e0-1ca692972787-image.png?alt=media" data-qx-storage-src="https://firebasestorage.googleapis.com/v0/b/quantrexacademy-app.firebasestorage.app/o/questions%2Ffigs%2Fpyq%2Fjee_main%2Fabb4563a-c619-42f1-ba79-d6cacc1c892a-05545eb3-cdc3-4eae-b4e0-1ca692972787-image.png?alt=media" style="width: 300px; height: 197px;" /> the graph of$ y = 2 + \cos x $for$ x \in \left[\pi, \infty\right) $<img alt="" src="/api/proxy-image?url=https%3A%2F%2Ffirebasestorage.googleapis.com%2Fv0%2Fb%2Fquantrexacademy-app.firebasestorage.app%2Fo%2Fquestions%252Ffigs%252Fpyq%252Fjee_main%252F86b84935-fd96-4bf4-a1b6-0ce1cea41692-d161592b-65a8-4598-8eb8-63d94282451b-image.png%3Falt%3Dmedia&clean=1&v=wm2" data-qx-orig-src="https://firebasestorage.googleapis.com/v0/b/quantrexacademy-app.firebasestorage.app/o/questions%2Ffigs%2Fpyq%2Fjee_main%2F86b84935-fd96-4bf4-a1b6-0ce1cea41692-d161592b-65a8-4598-8eb8-63d94282451b-image.png?alt=media" data-qx-storage-src="https://firebasestorage.googleapis.com/v0/b/quantrexacademy-app.firebasestorage.app/o/questions%2Ffigs%2Fpyq%2Fjee_main%2F86b84935-fd96-4bf4-a1b6-0ce1cea41692-d161592b-65a8-4598-8eb8-63d94282451b-image.png?alt=media" style="width: 300px; height: 240px;" />So graph of$ f\left(x\right) =\left\{ $\begin{cases}\sin x, & 0 \le x \le \dfrac{\pi}{2} \\ 1, & \dfrac{\pi}{2} \le x \le \pi \\ 2 + \cos x, & x > \pi\end{cases}$ \right)

$Let f: [0, ∞) → [0, 3] be a function defined by f(x) = \ array ll max\ t: 0 ≤ t ≤ π\ , x ∈ [0, π] 2 + x, x > π array ).$

$f$ \left(x\right) $is differentiable everywhere in \left(0, \infty\right)$

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