If P ( x ) be a polynomial with real coefficients such that P ( ² x )= P ( ² x ) , for all x [0, / 2] . Consider the following statements : I. P ( x ) is an even function. II. P ( x ) can be expressed as a polynomial in (2 x -1)² I. P ( x )

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If P ( x ) be a polynomial with real coefficients such that P ( ² x )= P ( ² x ) , for all x [0, / 2] . Consider the following statements : I. P ( x ) is an even function. II. P ( x ) can be expressed as a polynomial in (2 x -1)² I. P ( x ) is a polynomial of even degree Then,

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Correct answer: C. only II and III are true

array l P ( ² x )= P ( ² x ) P ( ² x )= P (1- ² x ) P ( x )= P (1- x ) x [0,1] array Differentiable both sides w.r.t. xP^ (x)=-P^ (1-x) So P^ (x) is symmetric about point x= 1 2 So P ^ ( x ) has highest degree odd P ( x ) has highest degree even

If P ( x ) be a polynomial with real coefficients such that P ( ² x )= P ( ² x ) , for all x [0, / 2] . Consider the following statements : I. P ( x ) is an even function. II. P ( x ) can be expressed as a polynomial in (2 x -1)² I. P ( x ) is a polynomial of even degree Then,

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