I recently came across an interesting expression:
$$f = \frac{X\cdot b\cdot\left(\frac{a}{a+b}\right)^X}{a}$$
Where we have the following constraints: $0 < (a,b) < 1$, $(a + b) \leq 1$, and $X > 1$. Taking the partial derivatives for $a$ and $b$, we find a critical point at $b = \frac{a}{X-1}$ under the constraint that $0 < a \leq \frac{X-1}{X}$, which satisfies the earlier constraint that $(a+b)\leq 1$.
Looking to confirm numerical evidence that this represents a maximum over the constrained interval, the second (mixed partial) derivative test was applied, but failed, and for an interesting reason. Setting $b = \frac{a}{X-1}$ we can reformulate $f$ as:
$f' = \left(\frac{X-1}{X} \right)^{X-1}$, where $a$ and $b$ vanish.
Can we prove that $f'$ represents a hard upperbound for $f$ under the specified constraints? How does one proceed under these circumstances?
