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KVPY2012MathematicsFunctions

Let f: R R be the function f(x)= (x-a₁ ) (x-a₂ )+ (x-a₂ ) (x-a₃ )+ (x-a₃ ) (x-a₁ ) with a₁, a₂, a₃ R . Then f(x) 0 if and only if -

Options

  1. AAt least two of a_ i , a₇, a₇ are equal
  2. Ba_ i =a_ i =a₇
  3. Ca ₁, a ₂, a ₃ are all distinct
  4. Da ₁, a ₂, a ₃ , are all positive and distinct

Correct answer

B. a_ i =a_ i =a₇

Step-by-step solution

Only when a ₁= a ₂= a ₃ In other cases f ( x ) will take both positive and negative values

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