Quantrex Quantrex AcademyJEE · NEET · NDA PYQs with solutions Open app
JEE Main202313 Apr 2023Morning ShiftMathematicsApplication of DerivativesActual

The set of all a ∈ ℝ for which the equation x | x - 1 | + | x + 2 | + a = 0 has exactly one real root, is

Options

  1. A- 7 , ∞
  2. B- ∞ , ∞
  3. C- 6 , - 3
  4. D- ∞ , - 3

Correct answer

B. - ∞ , ∞

Step-by-step solution

Given, x | x - 1 | + | x + 2 | + a = 0 Now taking, Case I: x < - 2 , we get - x 2 + x - x - 2 + a = 0 ⇒ a = x 2 + 2 And y = x 2 + 2 is decreasing ∀ x ∈ ( - ∞ , - 2 ) Now taking Case II: - 2 ≤ x < 1 we get, - x 2 + x + x + 2 + a = 0 ⇒ a = x 2 - 2 x - 2 And y = x 2 - 2 x - 2 So, d y d x = 2 x - 1 ≤ 0   ∀   x ∈ [ - 2 , 1 ) Hence, y is decreasing ∀ x ∈ [ - 2 , 1 ) Now taking Case III: x ≥ 1 we get, x 2 - x + x + 2 + a = 0 ⇒

Practice Application of Derivatives on Quantrex Academy →

More from Application of Derivatives

Consider the function f: (0, ) (- , ) given by f(x) = x , _e(x) - x + 1 . Then which one of the following statements is TRUE? 2026Let P be the point on the parabola y = x^2 such that the slope of the tangent to the parabola at the point P is 4 . Let Q be the point in the first quadrant lying on the circle x^2 2026Let f: R R be a differentiable function such that f ( x+y 3 ) = f(x) + f(y) 3 for all x, y R , and f'(0) = 3 . Then the minimum value of the function g(x) = 3 + e^x f(x) , is: 2026_ 0 x (16 ( x 2 ) ^3 ( x 2 ) ) is equal to: 2026Let f(x) be a polynomial of degree 5 , and have extrema at x = 1 and x = -1 . If _ x 0 ( f(x) x^3 ) = -5 , then f(2) - f(-2) is equal to: 2026The number of critical points of the function f(x) = cases | x x |, & x 0 1, & x = 0 cases in the interval (-2 , 2 ) is equal to : 2026Let f be a differentiable function satisfying f(x)=1-2 x+ ₀^ x e ^ (x-t) f(t) dt , x R and let g (x)= ₀^ x (f( t )+2)¹⁵( t -4)⁶( t +12)¹⁷ dt , x R . If p and q are respectively the 2026Consider the following three statements for the function f:(0, ) R defined by f(x)= | _ e x |-|x-1| : (I) f is differentiable at all x>0 . (II) f is increasing in (0,1) . (III) f i 2026 Full Application of Derivatives list All JEE Main PYQs