If we're looking for a local maximum, then it's clear that
\[d^2f >0\]
for any direction we choose, corresponding to any possible choice in the coefficients above, so
\[c_i \text{ are completely arbitrary} \]
Let's say eigenvalue k is not positive. Then, it's clear that if we chose
\[c_i = \delta_{i,k} \]
then our condition would not be satisfied. Therefore, for a point to be a local maximum, the eigenvalues of the Hessian matrix must all be positive. Since the determinant of any matrix is the product of its eigenvalues, obviously
\[det(H) > 0 \]
Similarly, if we seek a local minimum, all of the eigenvalues must be negative. In that case,
\[det(H) >0 \text{ if the space has even dimension, and }det(H)<0 \text{ if the space}\] \[\text{ has odd dimension} \]
if we have a saddle point, then d^2f changes sign depending on our direction. That means that some of the eigenvalues must be positive and some must be negative.