Solution
= Solution
The <stationary point> equations are $ae^x=c$ and $be^y=d$. The sign assumptions make both ratios positive, so each equation has exactly one real solution. Therefore
$$
\boxed{(x_*,y_*)=\left(\log\frac ca,\ \log\frac db\right)}.
$$
Substitution gives the stationary value
$$
\boxed{h_*=c+d-c\log\frac ca-d\log\frac db}.
$$