Let the function, f(x)= cases -3 a x^2-2, & x 1 a^2+b x, & x ≥ 1 cases be differentiable for all x ∈ R , where a 1, b ∈ R . If the area of the region enclosed by y=f(x) and the line y=-20 is α+β 3 , α, β ∈ Z , then the

Correct answer: 0 f(x) is continuous and differentiable aligned & at x=1, LHL=RHL, LHD=RHD & -3 a-2=a^2+b,-6 a=b & a=2 ; b=-12 & f(x)=\ array cc -6 x^2-2, & x < 1 4-12 x, & x ≥ 1 array . aligned aligned & . Area =∫_ - 3 ^1(-6 x^2-2+20) d x+∫_1^2(4-12 x+20) d x] & =16+12 3 +6=22+12 3 & α+β=34 aligned

JEE Main 202522 JanuaryMorning Shift

Exam: JEE Main
Year: 2025
Date: 22 January
Shift: Morning Shift
Subject: Mathematics
Chapter: Continuity and Differentiability

JEE Main 2025 · 22 January · Morning Shift · Mathematics · Continuity and Differentiability

Let the function, f(x)= cases -3 a x^2-2, & x 1 a^2+b x, & x ≥ 1...

Let the function,

f(x)=

\begin{cases}-3 a x^2-2, & x \lt 1 \\ a^2+b x, & x \geqslant 1\end{cases}$

be differentiable for all

x \in \mathbf{R}

, where

\mathbf{a}\gt1, \mathbf{b} \in \mathbf{R}

. If the area of the region enclosed by

y=f(x)

and the line

y=-20

is

\alpha+\beta \sqrt{3}, \alpha, \beta \in Z

, then the value of

\alpha+\beta$ is ________

Correct answer

Step-by-step solution

f(x)

is continuous and differentiable

\begin{aligned} & \text{ at } x=1, \mathrm{LHL}=\mathrm{RHL}, \mathrm{LHD}=\mathrm{RHD} \\ & -3 a-2=a^2+b,-6 a=b \\ & a=2 ; b=-12 \\ & f(x)=\left\{\begin{array}{cc} -6 x^2-2, & x < 1 \\ 4-12 x, & x \geq 1 \end{array} \right. \end{aligned}

\begin{aligned} & \left.\text{ Area }=\int_{-\sqrt{3}}^1\left(-6 x^2-2+20\right) d x+\int_1^2(4-12 x+20) d x\right] \\ & =16+12 \sqrt{3}+6=22+12 \sqrt{3} \\ & \therefore \quad \alpha+\beta=34\end{aligned}

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 Main PYQs