The textbook has the following theorem:
Suppose $f(x,y)$ and $g(x,y)$ have continuous partial derivatives in a domain $D\subset \mathbb R^2$, and that $(x^*,y^*)$ is both an interior point of $D$ and a local extreme point for $f(x,y)$ subject to the constraint $g(x,y)=c$. Suppose that $g_x(x^*,y^*)\ne0$ and $g_y(x^*,y^*)\ne0$, where $g_z$ is the partial derivative of $g$ wrt $z$. Then there exists a unique number $\lambda$ such that $$ \mathcal L(x,y)=f(x,y)-\lambda(g(x,y)-c) $$ has a stationary point at $(x^*,y^*)$.
To caution against the uncritical use of the Lagrange method, the book gives the following example: $$ \max_{x,y}2x+3y\quad \text{subject to}\quad \sqrt x+\sqrt y=5 $$ The Lagrange method produces $(9,4)$, whereas the global maximum is $(0,25)$. It then asks which assumption of the theorem is violated?
To me, the problem doesn't violate any assumption of the theorem. The partials exist and are continuous on $(0,\infty)$, $(9,4)$ is a local extremum, and the partials of $g$ are non-zero at $(9,4)$.
Is the point of the question simply to warn people not to confuse stationary points with global extrema? Or am I missing something here?
There is nothing wrong here, no assumption is violated, and Lagrange's method has produced exactly what it is supposed to do: The point $(9,4)$ is a conditionally stationary point of $f$ allright, and there are no other conditionally stationary points in the "relative interior" of the feasible set $$S:=\bigl\{(x,y)\>\bigm|\>x\geq0, \> y\geq0, \>\sqrt{x}+\sqrt{y}=5\bigr\}\ .$$ But the point $(9,4)$ does not solve the global maximum problem ${\cal P}$ at stake. The feasible set $S$ is a certain arc connecting the points $(0,25)$ and $(25,0)$. We now have a candidate list consisting of three points: The point $(9,4)$ in the interior of the arc, and the two end points. The global maximum is at one of these three points, and a comparison of values shows that this is the point $(0,25)$.
As an aside note that the assumption $g_x(x^*,y^*)\ne0$ and $g_y(x^*,y^*)\ne0$ is too strong; $\bigl(g_x(x^*,y^*),g_y(x^*,y^*)\bigr)\ne(0,0)$ is sufficient.