Let f(x)=(1−x)2sin2x+x2 for all x∈R, and let g(x)=x∫1(2(t−1)t+1−lnt)f(t)dt for all x∈(1,∞)Consider the statements:P:There exists some x∈IR such that f(x)+2x=2(1+x2)Q:There exists some x∈IR such that 2f(x)+1=2x(1+x)Then

Let f(x)=(1−x)2sin2x+x2 for all x∈R, and let g(x)=x∫1(2(t−1)t+

1.	Let f(x)=(1−x)2sin2x+x2 for all x∈R, and let g(x)=x∫1(2(t−1)t+1−lnt)f(t)dt for all x∈(1,∞)Consider the statements:P:There exists some x∈IR such that f(x)+2x=2(1+x2)Q:There exists some x∈IR such that 2f(x)+1=2x(1+x)Then
Answer» Let $f (x) = (1 - x)^{2} {sin}^{2} x + x^{2}$ for all $x \in R,$ and let $g (x) = x \int 1 (\frac{2 (t - 1)}{t + 1} - ln t) f (t) d t$ for all $x \in (1, \infty)$ Consider the statements: $P : There exists some x \in I R such that f (x) + 2 x = 2 (1 + x^{2})$ $Q : There exists some x \in I R such that 2 f (x) + 1 = 2 x (1 + x)$ Then