Quantrex Quantrex AcademyJEE · NEET · NDA PYQs with solutions Open app
AP EAMCET202418 May 2024Morning ShiftMathematicsApplication of DerivativesActual

The set of all real values of ' a ' such that the real valued function f(x)=x^3+2 a x^2+3(a+1) x+5 is strictly increasing in its entire domain is

Options

  1. A(- ,- 3 4 ) (3, )
  2. B(- 3 4 , 3 )
  3. C(1,3)
  4. D(- , 1) (3, )

Correct answer

B. (- 3 4 , 3 )

Step-by-step solution

Given f(x)=x^3+2 a x^2+3(a+1) x+5 Now, f^ (x)=3 x^2+4 a x+3(a+1) for strictly increasing of f(x) in its entire domain aligned & (4 a)^2-4 3 3(a+1) 0 & 16 a^2-36(a+1) 0 aligned 4 a^2-9 a-9 0 (4 a+3)(a-3) 0 So, a ( -3 4 , 3 )

Practice Application of Derivatives on Quantrex Academy →

More from Application of Derivatives

Consider the quadratic equation a x^2+b x+c=0 , where 2 a+3 b+6 c=0 and let g(x)= a x^3 3 + b x^2 2 +c x . Statement-I : The given quadratic equation ax ^2+ bx + c =0 has at least 2025The difference between the absolute maximum and absolute minimum values of the function f(x)=2 x^3-15 x^2+36 x-30 on [-1,4] is 2025If f(x)=x e^ x(1-x) , x R , then f(x) is 2025The angle between the curves y ^2= x and x ^2= y at the point (1,1) is 2025If the tangent of the curve 4 y^3=3 a x^2+x^3 drawn at the point (a, a) forms a triangle of area 25 24 sq.units with the coordinate axes then a = 2025If the function f(x)= x- ^2 x is defined on the interval [- , ] , then f is strictly increasing in the interval 2025If the Lagrange's mean value theorem is applied to the function f(x)=e^x defined on the interval [1,2] and the value of c (1,2) is k , then e^ k-1 = 2025If the tangent to the curve x y+a x+b y=0 at (1,1) makes an angle Tan ⁻¹ 2 with X -axis, then ab a + b = 2025 Full Application of Derivatives list All AP EAMCET PYQs