Find x and y that optimise
\begin{align} f(x,y) &= (-a-y)(\Psi(y)-\Psi(x+y)) + (b-x)(\Psi(x)-\Psi(x+y)) \\ &-\log \Gamma(x+y) + \log\Gamma(x) + \log\Gamma(y) \end{align}
where a, b are strictly positive constant.
Taking the derivative of $f(x,y)$ w.r.t. $x$, we get \begin{align*} \frac{\partial}{\partial x}f(x,y) = (b-x)\Psi'(x) - (b-x-a-y)\Psi'(x+y) \end{align*}
Taking the derivative of $f(x,y)$ w.r.t. $y$, we get \begin{align*} \frac{\partial}{\partial x}f(x,y) = (-a-y)\Psi'(y)-(b-x-a-y)\Psi'(x+y) \end{align*}
It gives $x = b$ and $y = -a$ which means $y<0$ since $a>0$. Here $\Psi$ is digamma and $\Psi'$ is trigamma.
How can I proceed from here to optimise $f(x,y)$ such that $x > 0$ and $y > 0$.?