<s>[INST] Minimizing $f(x)=A^{\frac{tx-1}{x-1}} \left( c^x \frac{\Gamma(0.5+x)}{\sqrt{\pi}} \right)^{\frac{1-t}{x-1}}$ subject to the constraint Let $f(r)$ be a function defined as follows 
\begin{align}
f(x)=A^{\frac{tx-1}{x-1}} \left( c^x \frac{\Gamma(0.5+x)}{\sqrt{\pi}} \right)^{\frac{1-t}{x-1}}
\end{align}
where $0 < A,c$ and $ t\in (0,1)$.
I want to solve 
\begin{align}
\min_{\frac{1}{t}<x}  f(x)
\end{align}
My approach: 
Let $g(x)=\ln(f(x))$ then both $g(x)$ and $f(x)$ have the same minimum where
\begin{align}
g(x)=\frac{tx-1}{x-1} \ln(A)+\frac{(1-t)x}{x-1} \ln(c)+\frac{1-t}{x-1} \ln \left(\Gamma(0.5+x) \right)-\frac{1-t}{2(x-1)}\ln(\pi)
\end{align}
and 
\begin{align}
&g'(x)=\\
&=\frac{1-t}{(x-1)^2} \ln(A)-\frac{1-t}{(x-1)^2} \ln(c)+(1-t)\frac{(x-1)\psi^{(0)}(x+0.5)-\log(\Gamma(x+0.5))}{(x-1)^2}+\frac{1-t}{2(x-1)^2}\ln(\pi)\\
&=\frac{1-t}{(x-1)^2} \left(\ln \left( \frac{ \sqrt{\pi} A}{c\Gamma(x+0.5)} \right)+ (x-1)\psi^{(0)}(x+0.5)\right)
\end{align}
where $\psi(x)$ is the so called digamma function. 
So, this means we have to focus on 
\begin{align}
h(x)=\ln \left( \frac{ \sqrt{\pi} A}{c\Gamma(x+0.5)} \right)+ (x-1)\psi^{(0)}(x+0.5)
\end{align}
for $\frac{1}{t} <x$.
But how to solve $h(x)=0$ or say for what $x$ is $h(x)>0$ ???
If this impossible to do then the approximated solution is also fine?
I also feel that there might be a simpler approach with out using derivative.
Thank you for any help.
 [/INST] Hopefully the following can somehow help or give some idea.
Rewrite
\begin{align}
h(x)=\ln \left( \frac{ \sqrt{\pi} A}{c\Gamma(x+0.5)} \right)+ (x-1)\psi^{(0)}(x+0.5)
\end{align}
as
\begin{align}
h(x)= (x-1)\psi^{(0)}(x+0.5)-
\ln \left( \Gamma(x+0.5) \right) +\alpha
\end{align}
where $\alpha =\ln \left( \frac{ \sqrt{\pi} A}{c} \right)$.
Plot using Mathematica


*

*graph of $h(x)$ intersection with $0$-plane
Plot3D[{[Alpha] + (x - 1) PolyGamma[0, 1/2 + x] - 
   Log[Gamma[1/2 + x]], 0}, {x, 0, 3}, {[Alpha], -2, 2}]



*Contour plot of $h(x)=0$ 
ContourPlot[{[Alpha] + (x - 1) PolyGamma[0, 1/2 + x] - 
       Log[Gamma[1/2 + x]], 0}, {x, 0, 5}, {[Alpha], -3, 4}]

</s>