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AP EAMCET2005MathematicsApplication of Derivatives

If x is real, then the minimum value of x^2-x+1 x^2+x+1 , is

Options

  1. A1 3
  2. B3
  3. C1 2
  4. D2

Correct answer

D. 2

Step-by-step solution

On differentiating w.r.t. x , we get f^ (x)= (x^2+x+1 )(2 x-1)- (x^2-x+1 )(2 x+1) (x^2+x+1 )^2 for maximum or minimum, put f^ (x)=0 array lc & (x^2+x+1 )(2 x-1)- (x^2-x+1 )(2 x+1)=0 & x^2+x-1- (-x^2+x+1 )=0 & 2 x^2-2=0 x= 1 array Now, f^ (c)= 2 x^2-2 (x^2+x+1 )^2 Again differentiating, we get aligned & (x^2+x+1 )^2(4 x)- (2 x^2-2 ) & f^ (x)= 2 (x^2+x+1 )(2 x+1) (x^2+x+1 )^4 aligned at x=1, f^ (x)>0 Therefore it is minimum at x=1 Put x=1 in equation (i), we get f(1)= 1-1+1 1+1+1 = 1 3 The minimum value is 1 3 .

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