Positive solution for an exponential equation

Question

Define a function of $t$, $F(t) = e^{X_1t}-e^{X_2t}-e^{X_3t}+e^{X_4t},$ for some fixed real values $X_1, X_2, X_3, X_4 \in \mathbb{R}$ and $X_1<X_2 < X_3 < X_4$. Whether $F(t)$ has at most one positive solution or not.

Answer 1

\begin{align} f(t) &= \exp(x_1t)-\exp(x_2t)-\exp(x_3t)+\exp(x_4t) \tag{1}\label{1} ,\\ &x_1, x_2, x_3, x_4 \in \mathbb{R} \text{ and } x_1 < x_2 < x_3 < x_4 . \end{align}

Just one of the suitable examples:

let $x_1=-\tfrac2{100},x_2=0,x_3=\tfrac1{100},x_4=\tfrac2{100}$ and $y=\exp(\tfrac t{100})$.

Then \eqref{1} is equivalent to equation \begin{align} y^{-2}-y^0-y+y^2&=0 ,\\ y^4-y^3-y^2+1&=0 ,\\ y^3(y-1)-(y+1)(y-1)&=0 ,\\ (y-1)(y^3-y-1)&=0 \tag{2}\label{2} . \end{align}

Ignoring the solution $y=1$, that is, $t=0$, since $f(0)=0$ is true for any $x_1, x_2, x_3, x_4 \in \mathbb{R}$, the cubic term $y^3-y-1$ in \eqref{2} has only one real root

\begin{align} y_r&=\tfrac16\sqrt[3]{108+12\sqrt{69}} +\frac{2}{\sqrt[3]{108+12\sqrt{69}}} ,\\ t_r&=100\ln y_r\approx 28.1 . \end{align}