tmua.me Tau Paper 2 · Question 1

For $-1\le x\le2$ and real $y$ , define

H(x,y)=(x^2-2)^2+(x-y)^2.

For a real number $k$ , consider the statements

\begin{aligned} P:&\ \text{For every }x\in[-1,2]\text{ there is a real }y\text{ with }H(x,y)<k.\\ Q:&\ \text{There is a real }y\text{ such that }H(x,y)<k\text{ for every }x\in[-1,2]. \end{aligned}

Which one of the following describes exactly the values of $k$ for which $P$ is true and $Q$ is false?

4<k\le5

4<k<\dfrac{3757}{729}

4\le k\le\dfrac{3757}{729}

4<k\le\dfrac{3757}{729}

4<k\le8

examday