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KCET2022MathematicsApplication of Derivatives

The function f(x)= (1+x)- 2 x 2+x is increasing on

Options

  1. A(- , )
  2. B( ,-1)
  3. C(-1, )
  4. D(- , 0)

Correct answer

C. (-1, )

Step-by-step solution

Given, f(x)= (1+x)- 2 x 2+x Differentiating the function w.r.t. x , we get aligned f^ (x) & = 1 (1+x) (0+1)-2 [ (2+x) 1-x(0+1) (2+x)^2 ] & = 1 1+x -2 [ 2+x-x (2+x)^2 ]= 1 1+x - 4 (2+x)^2 & = (2+x)^2-4(1+x) (1+x)(2+x)^2 = 4+x^2+4 x-4-4 x (x+1)(x+2)^2 & = x^2 (x+1)(x+2)^2 = ( x x+2 )^2 1 (x+1) aligned Sign of f^ (x) depends on the sign of 1 (x+1) aligned & f^ (x)>0 when x>-1 increasing & f^ (x) < 0 when x < -1 decreasing aligned Hence, f(x) is increasing on (-1, ) .

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