Let f : R → R be the function f ( x ) = ( x - a _1) ( x - a_ 2) + ( x - a_ 2) ( x- a_ 3) + ( x - a _3) ( x - a _1)with a_1, a_2, a_3 \in R. Then f (x) \geq 0 if and only if

at least two of a_1, a_2, a_3 are equal

a_1, a_2, a_3 are all positive and distinct

KVPY SX SB Exam 2012 Question Paper

Question : 11 of 120

Marks: +1, -0

Let f : R → R be the function

f ( x ) = ( x - a _1) ( x - a_ 2) + ( x - a_ 2) ( x- a_ 3) + ( x - a _3) ( x - a _1)

with

a_1, a_2, a_3 ∈ R

. Then

f (x) ≥ 0

if and only if

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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