Consider the following optimization problem $\max f(x,y):=a x - b x^2 + y - cy^2 $ under $g(x,y)= x + y + Z =0$.
Using Lagrange multipliers I can rewrite this into
$\max h(x,y):= f (x,y) + \lambda g(x,y)$.
Using Mathematica I get the optimal solution for $x$ to be $\dfrac{-1 + a + 2cZ}{2(b+c)}$, which is clearly increasing in $Z$.
Normally, I would use the implicit function theorem, i.e., $\frac{\partial x}{\partial Z}= \frac{\frac{\partial^2 h(x,y)}{(\partial Z)^2}}{\frac{\partial^2 h(x,y)}{\partial Z \partial x}}$. However, this leads to a division by $0$ and so the theorem is not applicable.
Question: Is there an alternative to get the sensitivity of $x$ with regard to $Z$ (or with regard to $a$) directly; specifically without calculating $\lambda$?