Solving optimization problems using derivatives and critical points I have a homework question which I have completed 2/3 of; however I am stuck on the last part of the question.
The question is: 

A drug used to treat cancer is effective at low doses with an efficacy that increases with the quantity of the drug. However, at sufficiently high doses, the drug becomes lethal. For positive values of the constants k_1 and k_2, the fraction of patients surviving cancer with this drug treatment is given by
  $$S(q)=\frac{q^2}{k_1^2+q^2}-\frac{q^2}{k_2^2+q^2}.$$

There are three parts to the question. I have answered the first two parts, but I don't know how to approach the last part, which in this case is question 3.

1) Find $S(0)$ and $\lim_{q\to\infty}S(q)$ and in each case explain what your findings mean in medical terms.

For part 1 I made $s(q)$ equal to $0$, which gave me
$$\frac 0 {k_1^2} - \frac 0 {k_2^2} = 0-0 = 0.$$
This indicates that when $q$ is equal to $0$, no patients have survived. I also used the limits for when $q$ is equal to $+\infty$; this gave me $(1-1)$, which is also equal to $0$. Therefore the conclusion here is that when $q$ is very high, hence infinity, the patients will not survive either.

2) What is the optimal daily drug quantity to administer in terms of $k_1$ and $k_2$?

For part two, I found the derivative of $s(q)$ and made is equal to $0$. So I found $s'(q)=0$.
This was a very long process which will take me forever to type here, however my final answer here was $q=\sqrt{k_2k_1}$, where $k_2$ and $k_1$ are constants. I am not sure if this is correct, but it looks pretty right to me.

3) Suppose Health Canada has only approved the use of the drug of up to 45 mg/day and suppose $k_1=25 \mathrm{mg}/\mathrm{day}$ is the same for all patients but $k_2$ varies from patient to patient. To determine a personalized treatment strategy it would be useful for physicians to have a plot of the optimal daily drug quantity as a function of $k_2$, call it $q∗(k_2)$. Sketch a plot of $q∗(k_2)$ and explain why you drew it that way. Hint: don't forget that $k_1<k_2$!

Tthis is the question that I need help approaching, any one care to help please? Am I meant to find the critical points..? Isn't that what I found in Part two? The graph it? If $k_2$ is meant to be greater than $k_1$ and $k_1$ is 25mg and overall it is 45mg, should $k_2$ be between 0 and 20mg?
Is this how my graph should be? 

Your calculation of the optimal $q$ is correct. 
The rest is easy. As a function of $k_2$, we have $q(k_2)=\sqrt{k_1k_2}=5\sqrt{k_2}$, at least for $25\lt k_2\lt 45$. This is not difficult to plot: the curve $y=5\sqrt{x}$ has a familiar shape. Our curve is (part of) a rightward opening parabola. 
