In this example, which of these vectors are support vectors?

Question

The hyperplane of hard margin SVM with $\phi$ kernel is calculated as following that input space using $\phi$ to map to higher dimension space.

$$f(\phi(x))=4\phi_1(x)+9\phi_2(x)+4\phi_3(x)$$

$$ \phi(x)=\left[ \begin{array}{c} \phi_1(x)=x_1^2\\ \phi_2(x)=x_2^2\\\phi_3(x)=x_1x_2\\\phi_4(x)=-x_1 \end{array} \right] $$

Which of the following can be a support vector?

$$ x=\left[ \begin{array}{c} +1\\ 1 \end{array} \right] $$ $$ y=\left[ \begin{array}{c} 1\\ -1 \end{array} \right] $$

Options:

1. $\ x=$ YES, $\ y=$ No

2. $\ x=$ YES, $\ y=$ YES

The correct answer is option $1$. My challenge is where is the trick in this question, because calculation by hand shown that none of them are support vectors? How can we choose an option here?

Answer 1

If 𝑓(𝜙(𝑥))=4𝜙1(𝑥)+9𝜙2(𝑥)+4𝜙3(𝑥) is the equation of the SVM, what in

the wikipedia entry for SVM is the formula $w^T x - b$, (or better $w^T \phi(x) - b$ after you use the kernel trick) then points in the support vector should have $f(\phi(x)) = \pm 1$ (see this figure in the wiki page).

So just plug in the values of the two points and check whether $f(\phi(x))$ and $f(\phi(y))$ is either +1 or -1.