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AP EAMCET202317 May 2023Morning ShiftMathematicsContinuity and DifferentiabilityActual

Let f(x) be a differentiable function such that f(1)=2 , f(2)=6 and f(x+y)=f(x)+k x y+ 4 3 y^2 x, y R then f(x)=

Options

  1. A4 x-2
  2. By-4 x^2+2 x-4
  3. C8 3 x^2+ 4 3
  4. D4 3 x^2+ 2 3

Correct answer

D. 4 3 x^2+ 2 3

Step-by-step solution

f(1)=2, f(2)=6f(x+y)=f(x)+k x y+ 4 3 y^2... (I) Put x=1 & y=1 aligned & f(1+1)=f(1)+k+ 4 3 6- 4 3 =2+k & k= 8 3 aligned Put x=1 and y=m-1 in equation (i), we get aligned & f(1+m-1)=f(1)+k 1(m-1)+ 4 3 (m-1)^2 & f(m)= 4 3 m^2+ 2 3 & f(x)= 4 3 x^2+ 2 3 aligned

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