I have the below function:
$$\pi(x) = \frac{s_0\cdot \left(1-\left(\frac{s_1}{s_1+x \cdot \lambda}\right)^{k}\right) \cdot r_1}{s_0\cdot \left(1-\left(\frac{s_1}{s_1+x \cdot \lambda}\right)^{k}\right) \cdot \gamma+ r_0}- x$$ with:
- $x, s_0, s_1, r_0, r_1 \in R^+$
- $\lambda, \gamma \in [0,1]$
- $k \in \{0.25, 4\}$
I am trying to find the $x$ that maximises $\pi(x)$ for both k values (by graphing the function on Desmos, one can see there exists a maximum).
I have tried to solve $\frac{d\pi(x)}{dx}=0$ but there is no root afaik.
What else should I try? How to find the maximum with a closed form formula?
EDIT: adding my reasoning on finding the root by solving $\frac{d\pi(x)}{dx}=0$, notations inspired from comments.
We solve $\frac{d \pi(x)}{dx} = 0$ i.e. $$ \frac{\left(s_{0}\cdot r_{1}\cdot r_{0}\cdot\lambda\cdot k\cdot\left(\frac{s_{1}}{s_{1}+\lambda\cdot x}\right)^{1+k}\right)}{s_{1}\cdot\left(s_{0}\cdot\gamma\cdot\left(\left(\frac{s_{1}}{s_{1}+\lambda\cdot x}\right)^{k}-1\right)-r_{0}\right)^{2}}-1=0$$
$$\left(s_{0}\cdot r_{1}\cdot r_{0}\cdot\lambda\cdot k\cdot\left(\frac{s_{1}}{s_{1}+\lambda\cdot x}\right)^{1+k}\right) = s_{1}\cdot\left(s_{0}\cdot\gamma\cdot\left(\left(\frac{s_{1}}{s_{1}+\lambda\cdot x}\right)^{k}-1\right)-r_{0}\right)^{2}$$
We introduce:
- $a=s_{0}\cdot r_{1}\cdot r_{0}\cdot\lambda \cdot k$
- $y=s_1+\lambda \cdot x$
and rewrite $$ \pm\sqrt{a} \cdot s_1^{\frac{k}{2}} \cdot y^{-\frac{k+1}{2}} = s_0 \cdot \gamma \cdot \left(\left(\frac{s_{1}}{y}\right)^{k}-1\right)-r_{0} $$ $$ \pm \sqrt{a} \cdot s_1^{\frac{k}{2}} \cdot y^{-\frac{k+1}{2}} - s_0 \cdot \gamma \cdot s_{1}^k \cdot y^{-k} + s_0 \cdot \gamma +r_{0} =0 $$
and going one step further:
$$A \cdot y^{-\frac{k+1}{2}} + B \cdot y^{-k} + C= 0$$ with:
- $A=\pm \sqrt{s_{0} \cdot r_{1} \cdot r_{0} \cdot \lambda \cdot k \cdot s_1^k}$
- $B = - s_0 \cdot \gamma \cdot s_{1}^k$
- $C=s_0 \cdot \gamma +r_{0}$
We still have $k \in \{0.25, 4\}$ which for me makes it difficult to solve.
Note: One comment states this could be solved as a general trinomial via series expansion, which I do not yet master, what is the approach to get there?