How can I prove the function $f(t) = t\left(s-1\right)+1-s^t$ is strictly positive for $t \in (0, 1)$ when $s \geq 0, s \neq 1$? I tried using inequalities but could not find a satisfactory solution.
Here is a plot of the relevant graph (in green) on Desmos: https://www.desmos.com/calculator/wqttkdfinw.
Hint: the case $s=0$ is trivial, otherwise note that $\,f(0)=f(1)=0\,$ and $f$ is strictly concave for $s \gt 0, s \ne 1$ since $f''(t) = - \ln^2(s) \cdot s^t \lt 0\,$.