Finding limit supremum

Question

While finding the limit supremum of $5x\cos(πx) - x^2$, the book says it is $-\infty$. But this function is bounded above. Shouldn't it's supremum be the lowest upper bound(which is 6). Why does it say it's supremum is $-\infty$ ?

Answer 1

Note that $f(x)=5x\cos(πx) - x^2$ is continuos and

• $\lim_{x\to \pm \infty} f(x)=-\infty$

then by EVT the function has maximum and is bounded from above.