Let f(x)=\ array lc 3 x, & x 0 \ 1+x+[x], x+2[x]\ , & 0 ≤ x ≤ 2 5, & x 2, array . where [.] denotes greatest integer function. If α and β are the number of points, where f is not continuous and is not differentiable

Correct answer: 0 f(x)=\ array ccc 3 x & ; & x 2 array . f(x)=\ array ccc 3 x & ; & x 2 array . Not continuous at x ∈\ 1,2\ α=2 Not diff. at x ∈\ 0,1,2\ β=3 α+β=5

JEE Main 202528 JanuaryMorning Shift

Exam: JEE Main
Year: 2025
Date: 28 January
Shift: Morning Shift
Subject: Mathematics
Chapter: Continuity and Differentiability

JEE Main 2025 · 28 January · Morning Shift · Mathematics · Continuity and Differentiability

Let f(x)=\ array lc 3 x, & x 0 \ 1+x+[x], x+2[x]\ , & 0 ≤ x ≤ 2 5,...

Let

\mathrm{f}(x)=\left\{

\begin{array}{lc}3 x, & x \lt 0 \\ \min \{1+x+[x], x+2[x]\}, & 0 \leq x \leq 2 \\ 5, & x\gt2,\end{array}

\right.

where [.] denotes greatest integer function. If

\alpha

and

\beta

are the number of points, where f is not continuous and is not differentiable, respectively, then

\alpha+\beta

equals __________ $Let f(x)=\ array lc 3 x, & x 0 \ 1+x+[x], x+2[x]\ , & 0 ≤ x ≤ 2 5, & x 2, array . where [.] denotes greatest integer$

Correct answer

Step-by-step solution

f(x)=\left\{

\begin{array}{ccc}3 x & ; & x < 0 \\ \min \{1+x, x\} & ; & 0 \leq x < 1 \\ \min \{2+x, x+2\} & ; & 1 \leq x < 2 \\ 5 & ; & x>2\end{array}

\right.

f(x)=\left\{

\begin{array}{ccc}3 x & ; & x < 0 \\ x & ; & 0 \leq x < 1 \\ x+2 & ; & 1 \leq x < 2 \\ 5 & ; & x>2\end{array}

\right.

$Let f(x)=\ array lc 3 x, & x 0 \ 1+x+[x], x+2[x]\ , & 0 ≤ x ≤ 2 5, & x 2, array . where [.] denotes greatest integer$
Not continuous at

\mathrm{x} \in\{1,2\} \Rightarrow \alpha=2

Not diff. at

x \in\{0,1,2\} \Rightarrow \beta=3

\alpha+\beta=5

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