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Most Important Selected Qs for JEE AdvancedMathematicsApplication of Derivatives

Let f:(0, ) R be a differentiable function satisfying the equation 2 f(x)=f(x)+f ( x y ) x, y>0 . If f(1)=0 and f^ (1)=1 , then: [Note: Where [k] denotes greatest integer function less than or equal to k ]

Options

  1. Af(x) has no local maxima and no local minima.
  2. B_ x 0⁺ [ f(x+1) x ]=0
  3. Cf(x)=e x has no roots.
  4. Dthe equation 2 e f(x)=x has one distinct solution.

Correct answer

C. f(x)=e x has no roots.

Step-by-step solution

Differentiate w.r.t. y keeping x constant 0=x f^ (x y)- x y^2 f^ ( x y ) 1 y^2 f^ ( x y )=f^ (x y) Put y=x aligned & f^ (1)=x^2 f^ (x^2 )=1 & f^ (t)= 1 t t>0 & f(x)= x aligned

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