I have the following function:
$$g(t) = c_1 \cdot e^{-4t}+c_2 \cdot e^{-t}$$
and the constants that multiply them are:
$$c_1 = {\frac {16+a_{{0}}-4\,a_{{1}}}{-4+b_{{0}}}}$$
$$c_2 = -{\frac {a_{{0}}-b_{{0}}a_{{1}}+{b_{{0}}}^{2}}{-4+b_{{0}}}}$$
Also, I know beforehand that $a_0 \in [1, 9]$, $a_1 \in [-6,-2]$ and $b_0 \in [1,3]$.
I would like to know the maximum value that $|g(t)|$ could reach (it is important to highlight that $t$ is time, therefore can only be non-negative).
Sorry if this is a very easy optimization problem, but I am not very used to these techniques. Thank you all!
Edit: I previously had a typo (double minus, which equals plus, before the 2nd term of the numerator of c2). That has now been corrected. My earlier comments were based on solving the problem which had the typo.
The max value of $|g(t)|$, as well as $g(t)$, is $15.347792$, and is achieved at $t = 0.564865, a_0 = 9, a_1 = -6, b_0 = 3$, which produces corresponding $c_1 = -49, c_2 = 36$.
How do I know? I ran the "rigorous" numerical global optimizer YALMIP's BMIBNB and some other checks.