Let f ( x ) = ( 1 − x ) 2 sin 2 x + x 2 for all x ∈ I R and let g ( x ) = ∫ 1 x ( t + 1 2 ( t − 1 ) − ln t ) f ( t ) d t for all x ∈ ( 1 , ∞ ) . Question: Consider the statements: P : There exists

Question

Let f ( x ) = ( 1 − x ) 2 sin 2 x + x 2 for all x ∈ I R and let g ( x ) = ∫ 1 x ​ ( t + 1 2 ( t − 1 ) ​ − ln t ) f ( t ) d t for all x ∈ ( 1 , ∞ ) . Question: Consider the statements: P : There exists some x ∈ R such that f ( x ) + 2 x = 2 ( 1 + x 2 ) Q : There exists some x ∈ R such that 2 f ( x ) + 1 = 2 x ( 1 + x ) Then

TestHub · Accepted Answer

For the statement P , f ( x ) + 2 x = 2 ( 1 + x 2 ) ⇒ ( 1 − x ) 2 sin 2 x + x 2 + 2 x = 2 ( 1 + x 2 ) ⇒ ( 1 − x ) 2 sin 2 x = x 2 − 2 x + 1 + 1 ⇒ ( 1 − x ) 2 sin 2 x = ( 1 − x ) 2 + 1 ⇒ ( 1 − x ) 2 cos 2 x = − 1 , which is not possible for any real value of x . Hence P is not true. Let H ( x ) = 2 f ( x ) + 1 − 2 x ( 1 + x ) H ( 0 ) = 2 f ( 0 ) + 1 − 0 = 1 and H ( 1 ) = 2 f ( 1 ) + 1 − 4 = − 3 Hence, H ( x ) has a solution in ( 0 , 1 ) Therefore, Q is true.

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