Let f(x)=(1-x)² ² x+x² for all x I R and let g(x)= ₁^ x ( 2(t-1) t+1 - 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+x² )Q : There exists some x R such that 2 f(x)+1=2 x(

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Let f(x)=(1-x)² ² x+x² for all x I R and let g(x)= ₁^ x ( 2(t-1) t+1 - 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+x² )Q : There exists some x R such that 2 f(x)+1=2 x(1+x) Then

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Correct answer: C. P is false and Q is true

For the statement P, f(x)+2 x=2 (1+x² ) (1-x)² ² x+x²+2 x=2 (1+x² ) (1-x)² ² x=x²-2 x+1+1 (1-x)² ² x=(1-x)²+1 (1-x)² 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.

Let f(x)=(1-x)² ² x+x² for all x I R and let g(x)= ₁^ x ( 2(t-1) t+1 - 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+x² )Q : There exists some x R such that 2 f(x)+1=2 x(1+x) Then

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