Let f(x) = log(1 + x^{ 2}) and A be a constant such that \frac{| \f(x)- \f(y) |}{| x -y |}\leqA for all x, y real and x ≠ y. Then the least possible value of A is

KVPY SX SB Exam 19-Nov-2017 Question Paper

Question : 90 of 120

Marks: +1, -0

Let f(x) = log(1 + x

^{ 2}

) and A be a constant such that

{| \f(x)- \f(y) |}/{| x -y |}≤A

for all x, y real and x ≠ y. Then the least possible value of A is

Solution:

\f(x) = {2x}/{1 + x ^{ 2}}

Range of

{2x}/{1 + x ^{ 2}}∈ [-1, 1]

{| \f(x)- \f(y) |}/{| x -y |}≤A

means
maximum value of

{| \f(x)- \f(y) |}/{| x -y |}

is alwaysless than or equal to A.
So, least value of A is 1

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