## Optimization Help; Due tomorrow...

For some reason, the assignment questions in my calculus class are always much harder than the examples they give to you in class. Frankly, I think I'm pretty good at optimization, but I just can't wrap my head around this one question. Help is much appreciated because this assignment is due tomorrow at 12:15PM. This is just a first year calculus question, so hopefully someone can help me.

As an epidemic spreads through a population, the number of infected people, I, is expressed as a function of the number if susceptible people, S, by

I = k ln(S/S0) - S + S0 + I0, for k, S0, I0 > 0.

a) find the maximum number of infected people.
b) The constant k is a characteristic of the particular disease; the constants S0 and I0 are the values of S and I when the disease starts. Which of the following affects the maximum possible value of I? Explain.
- The particular disease, but not how it starts
- How the disease starts, but not the particular disease.
- Both the particular disease and how it starts.
The 0 part of S0 and I0 are supposed to be subscripted, but I can't manage to do that for some reason. I'm sure I can get the b) part myself, it's just getting the maximum. Just hoping someone can help me out with this one.