Quantrex Quantrex AcademyJEE · NEET · NDA PYQs with solutions Open app
Most Important Selected Qs for JEE AdvancedMathematicsApplication of Derivatives

Let f^ (x) be a continuous function which maps from [0,1] [p(a), p(b)] . If p(x) is a differentiable function on [a, b] such that p(g(x))=x, g(0)=a and g(1)=b , then which of the following is/are true?

Options

  1. Af^ (0)+2 < f(1)
  2. Bf(1) 1+f(0)
  3. C₀^1 f^ (x) d x ₀^1 g^ (x) d x p^ (c) for some c (a, b)
  4. DThere exists k [0,1] such that f^ (k)=k

Correct answer

D. There exists k [0,1] such that f^ (k)=k

Step-by-step solution

First observe that g is the inverse of p . Hence p(a)=0, p(b)=1 . Also f^ is a continuous function which maps from [0,1] [0,1] by Intermediate value theorem we can say that there is k [0,1] such that f^ (k)=k ( d ) Now, f^ (x) 1 (p(b)-p(a)) Hence, ₀^1 f^ (x) d x (p(b)-p(a)) Divide both sides by b-a= ₀^1 g^ (x) d x ₀^1 f^ (x) d x ₀^1 g^ (x) d x (p(b)-p(a)) b-a =p^ (c), c (a, b) [ By LMVT] (c) Also, use LMVT for y=f(x) in [0,1] f^ (c)= f(1)-f(0) 1-0 1f(1)-f(0) 1f(1) 1+f(0) (b)

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 Most Important Selected Qs for JEE Advanced PYQs