KVPY 2019 SB SX Exam Previous Papers

Question : 17 of 120

Marks: +1, -0

Let f : R → R be a continuous function such that

\f(x^2) = \f(x^3)

for all x ∈ R. Consider the following statements
I. f is an odd function
II. f is an even function
III. f is differentiable everywhere
Then

Solution:

\f(x^2 ) = \f(x^3 )

⇒

\f(α) = f(α^{2\/3}) = \f(α^{4\/9} ) = ...... = \f( α^{ 2^n \/3 ^n} )

= f(α^0 ) \as n →∞

\f(a) = \f(1) ⇒ \f(x)

is a constant function ⇒ f(x) is differentiable and even function

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