tmua.me Extreme Set A Paper 2 · Question 1

Let $k\geq0$ . Consider the following statements.

S: For every real number $x$ with $0\leq x\leq1$ , there exists a real number $y$ such that

|y-x|\leq k(1+x)\qquad\text{and}\qquad |y-x^2|\leq k(1+x).

T: There exists a real number $y$ such that, for every real number $x$ with $0\leq x\leq1$ ,

|y-x|\leq k(1+x)\qquad\text{and}\qquad |y-x^2|\leq k(1+x).

For which values of $k$ is S true and T false?

\dfrac1{12}\leq k<\dfrac13

\dfrac{3-2\sqrt2}{4}\leq k<\dfrac13

\dfrac{3-2\sqrt2}{2}\leq k<\dfrac13

3-2\sqrt2\leq k<\dfrac13

\dfrac18\leq k<\dfrac13

\dfrac1{16}\leq k<\dfrac13

examday