I want to find the optimal solution to the following seemingly sum of two exponential functions:
$$\max_{x \in [0, 1]}~f(x) = \underbrace{x\exp\left(-a\left(bx^t+1\right)^\frac{1}{1-t}\right)}_{f_{1}(x)} + \underbrace{(1-x)\exp\left(-c(1-x)^{\frac{t}{1-t}}\right)}_{f_2(x)},$$ where $a, b, c > 0$ and $t \in (0, 1)$.
My attempt: I tried obtaining the optimal solution $x^*$ by exploring its properties. For example, I did the following: \begin{align}&\frac{\mathrm{d} f(x)}{\mathrm{d} x} = 0 \end{align} which results in $$\frac{\mathrm{d} f_1(x)}{\mathrm{d} x} = -\frac{\mathrm{d} f_2(x)}{\mathrm{d} x}.$$ Thus, taking the first derivatives of $f_1(x)$ and $f_2(x)$ with respect to $x$ yields $$ \dfrac{\left(\left(t-1\right)\left(bx^t+1\right)^\frac{t}{t-1}+abtx^t\right)\mathrm{e}^{-\frac{a}{\left(bx^t+1\right)^{1/(t-1)}}}}{\left(bx^t+1\right)^\frac{t}{t-1}} = \left(ct\left(1-x\right)^\frac{t}{1-t}+t-1\right)\mathrm{e}^{-c\left(1-x\right)^\frac{t}{1-t}}. $$ Now, the right hand side is positive and increasing only if $ct\left(1-x\right)^\frac{t}{1-t}+t-1 > 0$. But I cannot find a trend in the left hand side to say something about the uniqueness of the optimal solution $x^*$ and the conditions on the existence of $x^*$. How can one get the optimal solution or at least some properties regarding the optimal solution such as uniqueness and reasonable bounds on the optimal solution?