QJEEM23Ma1388: JEE Main 2023 Mathematics - Class 12 JEEM23Ma1388 Solution: Let denote the greatest integer function and . Let be the…

Question

Verified Solution • 2025–26 Session moderate

Let

[x]

denote the greatest integer function and

f(x)=\max \{1+x+[x], 2+x, x+2[x]\}, 0 \leq x \leq 2

. Let

m

be the number of

points in

[0,2]

, where

f

is not continuous and

n

be the number of points in

(0,2)

, where

f

is not differentiable. Then

(m+n)^{2}+2

is equal to :

Accepted Answer

\begin{aligned} Let g (x) = 1 + x + [x] = {\begin{array}{cc} 1 + x; & x \in [0, 1) \\ 2 + x; & x \in [1, 2) \\ 5; & x = 2 \end{array} \\ λ (x) = x + 2 [x] = {\begin{array}{cc} x; & x \in [0, 1) \\ x + 2; & x \in [1, 2) \\ 6; & x = 2 \end{array} \\ r (x) = 2 + x \\ f (x) = {\begin{array}{cc} 2 + x; & x \in [0, 2) \\ 6; & x = 2 \end{array} \end{aligned}

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

is discontinuous only at

x=2 \Rightarrow \mathrm{m}=1

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

is differentiable in

(0,2) \Rightarrow \mathrm{n}=0

(m + n)^{2} + 2 = 3

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