S=1000000*exp(0.000005*t).*sin(2*pi*5000*t.^2);
Consider sin(2*pi*5000*t): that would be a sine wave with frequency 5000 cycles per 2*pi, so with sin(2*pi*5000*t) you would expect 5000 cycles approximately every 6.3, or approximately 796 cycles per unit. But your t is 1:100 so you are sampling less than 1/796 th of the peaks per unit.
But you do not have sin(2*pi*5000*t) you have sin(2*pi*5000*t^2) . Does that increase or decrease the frequency? Clearly it increases the frequency -- by 75-ish your sine frequency is over 2 megahertz.
Where is the true minima?
Well, exp(0.000005*t) is exponential increasing in t, and sin() of real values stays in the range -1 to +1 so for any t that gives a positive S, a value of t that leads to exactly one cycle later in the sine wave would give the same sin() value but would have increasing exp(0.000005*t) (because larger t) and so would lead to a larger value of S. Likewise, for any given t that gives a positive S, there is a slightly larger value of t1 that is going to lead to the sin() being -1, and then there would be a t2 slightly larger than that which would lead to the same point on the sin() cycle but with t2>t1 then exp(0.000005*t2) > exp(0.000005*t1) and it follows that S for t2 would be more negative than the S for t1.
We thus see that no matter what minima we find for S, there is a time nearby that will give more of a minima. Therefore the true minima is arbitrarily small.
(In this case you would not say that the minima is -infinity at t = infinity because sin(infinity) is not defined.)
