Let f: [- (1)/(2), 2] → ℝ and g: [- (1)/(2), 2] → ℝ be functions defined by f(x) = [ x ^ 2 - 3] and g (x) = |x|f(x) + |4x - 7| f(x) , where [y] denotes the greatest integer less than or equal to y for y ∈ ℝ . Then

Correct answer: B. f is discontinuous exactly at four points in [- (1)/(2), 2] f(x) = [ x ^ 2 ] - 3 g(x) = (|x| + |4x - 7|)([ x ^ 2 ] - 3) ∵ f is discontinuous at (when x 2 is an integer) x = 1, 2 , 3 , 2in[- (1)/(2), 2] and |x| + |4x - 7| ≠ 0 at x = 1, 2 , 3 , 2 ⇒ g(x) is discontinuous at x = 1, 2 , 3 in (- (1)/(2), 2) In (0 - , 0 + ) ( is very small) ⇒ g(x) = (|x| + |4x - 7|) . (- 3) ⇒ 'g' is non derivable at x = 0 .as | x | is there which is not diff. at x = 0 In ((7)/(4) - , (7)/(4) + ) ⇒ g(x) = 0 as f(x) = 0 ⇒ Derivable at x = (7)/(4) ∴ 'g' is non - derivable at 0, 1, 2 , 3

JEE Advanced 2016Paper 2

Exam: JEE Advanced
Year: 2016
Paper: Paper 2
Subject: Mathematics
Chapter: Continuity and Differentiability

JEE Advanced 2016 · Paper 2 · Mathematics · Continuity and Differentiability

Let f: [- (1)/(2), 2] → ℝ and g: [- (1)/(2), 2] → ℝ be functions...

Let

f: \left[- \dfrac{1}{2}, 2\right] \to \mathbb{R} and

g: \left[- \dfrac{1}{2}, 2\right] \to \mathbb{R}

be functions defined by

f\left(x\right) = \left[{x}^{2} - 3\right]

and g \left(x\right) =

\left|x\right|f\left(x\right) + \left|4x - 7\right|

f\left(x\right)

, where \left[y\right]

denotes the greatest integer less than or equal to

for

y \in \mathbb{R} . Then

Options

Af $is discontinuous exactly at three points in \left[- \dfrac{1}{2}, 2\right]$
Bf $is discontinuous exactly at four points in \left[- \dfrac{1}{2}, 2\right]$
C $g$ is NOT differentiable exactly at four points in $\left(- \dfrac{1}{2}, 2\right)$
D $g$ is NOT differentiable exactly at five points in $\left(- \dfrac{1}{2}, 2\right)$

Correct answer

B. f $is discontinuous exactly at four points in \left[- \dfrac{1}{2}, 2\right]$

Step-by-step solution

f\left(x\right) = \left[{x}^{2}\right] - 3

g\left(x\right) = \left(\left|x\right| + \left|4x - 7\right|\right)\left(\left[{x}^{2}\right] - 3\right)

∵

f

is discontinuous at (when x² is an integer)

x = 1, \sqrt{2}, \sqrt{3}, 2in\left[- \dfrac{1}{2}, 2\right]

and \left|x\right| + \left|4x - 7\right| \ne 0

at

x = 1, \sqrt{2}, \sqrt{3}, 2

<br />

⇒ g\left(x\right)

is discontinuous at

x = 1, \sqrt{2}, \sqrt{3}

in \left(- \dfrac{1}{2}, 2\right)

In

\left(0 - \delta, 0 + \delta\right)

(

\delta

is very small)<br />

⇒ g\left(x\right) = \left(\left|x\right| + \left|4x - 7\right|\right) . \left(- 3\right)

<br />

⇒

'g' is non derivable at

x = 0

.as

| x |

is there which is not diff. at

x = 0

<br />

\left(\dfrac{7}{4} - \delta, \dfrac{7}{4} + \delta\right)

⇒ g\left(x\right) = 0

f\left(x\right) = 0

⇒

Derivable at

x = \dfrac{7}{4}

∴

'g' is non - derivable at

0, 1, \sqrt{2}, \sqrt{3}

Practice Continuity and Differentiability on Quantrex Academy →

More from Continuity and Differentiability

Let ℝ denote the set of all real numbers. Let f : ℝ arrow ℝ be an arbitrary function and… JEE Advanced 2026 Consider the function f : (-(π)/(2), (π)/(2)) arrow (-∞, ∞) defined by f(x) = (|x| + |x… JEE Advanced 2026 For a real number α , let [α] denote the greatest integer less than or equal to α . For a… JEE Advanced 2026 For the function f(x) = e^ |x| - |x| , x ∈ ℝ , consider the following statements… JEE Main 2026 Let f(x) = cases (1)/(3), & x ≤ π/2 (b(1- x))/((π-2x)^2), & x > π/2 cases . If f is… JEE Main 2026 Let f(x) = cases x^3 + 8 ; & x Then the number of points, where the function g ∘ f is… JEE Main 2026 Let f(x)= cases e^ x-1 , & x<0 x^2-5x+6, & x ≥ 0 cases and g(x)=f(|x|)+|f(x)| . If the… JEE Main 2026 The number of points, at which the function f(x) = \ 6x, 2 + 3x^2\ + |x - 1|| |x^2… JEE Main 2026 The number of points in the interval [2, 4] , at which the function f(x) = [x^2 - x… JEE Main 2026 Let f(x)= _ θ arrow 0 ( π x-x^ ((2)/(θ)) (x-1) 1+x^ ((2)/(θ)) (x-1) ), x ∈ R . Consider… JEE Main 2026 Let [t] denote the greatest integer less than or equal to t . If the function f(x)=\… JEE Main 2026 If the function f(x)= e^ x (e^ x-x -1)+ _ e ( x+ x)-x x-x is continuous at x=0 , then the… JEE Main 2026 Let α, β ∈ ℝ be such that the function f(x)= cases 2 α(x^ 2 -2)+2 β x &, x be… JEE Main 2026 If f(x)=\ array cc a|x|+x^ 2 -2( |x|)( |x|) x &, x ≠ 0 b &, x=0 array . is continuous at… JEE Main 2026 Let f(x)= cases a x^ 2 +2 a x+3 4 x^ 2 +4 x-3 &, x ≠-(3)/(2), (1)/(2) ~b &, x=-(3)/(2)… JEE Main 2026 Let [·] denote the greatest integer function, and let f(x)= \ 2 x, x^ 2 \ . Let S = \ x… JEE Main 2026 Full Continuity and Differentiability list Continuity and Differentiability JEE Maths PYQs JEE Mathematics PYQs All JEE Advanced PYQs