If you multiply a vector $x$ with a matrix $A$, where $x$ has unit Euclidean norm, is there any direction in which $x$ can point, which results in a smaller resulting magnitude than the smallest eigenvalue of $A$?
Meaning is the best choice for $x$ to minimize $\|Ax\|_2$ the eigenvector with the smallest eigenvalue?
$$ A = \begin{pmatrix} 3 & 6 & 2 & 6\\ 1 & 7 & 8 & 7\\ 1 & 6 & 5 & 9\\ 3 & 4 & 6 & 5 \end{pmatrix} $$
We can formulate your issue as:
Minimize $\|Ax\|$ under the constraint $\|x\|=1$
than can be re-phrased in the following equivalent way (because it is the Euclidean norm)
Minimize $\|Ax\|^2=(Ax)^T(Ax)=(x^TA^T)(Ax)=x^T(A^TA)x$ under the constraint $\|x\|=1$
As a consequence, it is the eigenvectors of $B:=A^TA$ that must be considered.
(have you heard about SVD and singular vectors and singular values ?)