= Solution
The <Hessian matrix> is diagonal:
$$
H(x,y)=\begin{pmatrix}ae^x&0\\0&be^y\end{pmatrix},\qquad
H(x_*,y_*)=\begin{pmatrix}c&0\\0&d\end{pmatrix}.
$$
Its <eigenvalues> are $c$ and $d$. Hence the <second-derivative test> gives a strict <local minimum> if $c,d>0$, a strict <local maximum> if $c,d<0$, and a <saddle point of a scalar function> if $cd<0$. No degenerate case occurs because both are nonzero.
The <local minimum> is the unique <global minimum>, since both separated summands are <strictly convex functions> and tend to $+\infty$ at either end of the real line. Similarly, when both signs are negative there is a unique <global maximum>.
Back to article page
