Lower and upper bound on real inner product with eigenvalues

Question

Assume $A \in R^n$ is a real positive definite symmetric matrix, then for any vector $x \in R^{n \times 1}$, we have
$$
\underline{\lambda} x^\top x \leq x^\top A x \leq \overline{\lambda} x^\top x
$$
where $\underline{\lambda} $ and $\overline{\lambda}$ are the minimum and maximum eigenvalues of $A$.
I would like to know if it is correct also, for any $x,y \in R^{n \times 1}$ that $x_i y_i > 0$, for $i = 1, …, n$, i.e.
$$
\underline{\lambda} x^\top y \leq x^\top A y \leq \overline{\lambda} x^\top y
$$

Answer 1

$x=(2,3)$, $y=(3,2)$, $A=\pmatrix{7&-1\cr-1&7\cr}$, eigenvalues $6,8$; $6x^ty=72$, $x^tAy=71$.