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MHT CET202522 Apr 2025Morning ShiftMathematicsApplication of DerivativesActual

If x is real, then the difference between the greatest and least values of x^2-x+1 x^2+x+1 is

Options

  1. A10 3
  2. B8 3
  3. C5 3
  4. D1 3

Correct answer

B. 8 3

Step-by-step solution

Let y = x^2 - x + 1 x^2 + x + 1 . To determine the range of y , we eliminate the denominator and rearrange: y(x^2 + x + 1) = x^2 - x + 1 yx^2 + yx + y = x^2 - x + 1 (y - 1)x^2 + (y + 1)x + (y - 1) = 0 For real x , the discriminant D of this quadratic must satisfy D 0 : D = (y + 1)^2 - 4(y - 1)^2 0 (y + 1)^2 - 4(y - 1)^2 0 (3 - y)(3y - 1) 0 This inequality holds for 1 3 y 3 , including y = 1 as x = 0 is valid. The range is [ 1 3 , 3] , so the greatest value is 3 and the least is 1 3 . The difference is: 3 - 1 3 = 8

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