<s>[INST] If $r = \min \left\{ a , \frac{b}{ a+ a^2 b^3} \right\}$ , find $ r_{\max}$ Studying differential equations I came cross through this:
Let $ \displaystyle{ r = \min \left\{ a , \frac{b}{ a+ a^2 b^3} \right\} } $, where $ a,b >0$. Find $ r_{ \max} $.
Here is what I did:
Fix $ a> 0$ and define $ \displaystyle{ g(b) = \frac{b}{ a+ a^2 b^3} ,\quad b>0 }$. Then we have that $ \displaystyle{ g'(b) = \frac{ a(1-2ab^3)}{ (a+ a^2 b^3)^2} }$.
Thus,  $g$ has maximum at $ \displaystyle{b_0= \frac{1}{ \sqrt[3]{2a} }}$ which gives $ \displaystyle{ g(b) \leq g\left( \frac{1}{ \sqrt[3]{2a}} \right) = \frac{2}{ 3a \sqrt[3]{2a} }}$
So, $ \displaystyle{ r = \min \{ a , \frac{2}{ 3a \sqrt[3]{2a} } \}}$. I think that to find $a$ which gives $ r_{\max}$ I have to solve the equation $ \displaystyle{ a=\frac{2}{ 3a \sqrt[3]{2a}} }$.
Is this correct and why....?
Any ideas?
Thank's in advance!
edit: I edit the title.
 [/INST] The maximum is when $$a=(4/27)^{1/7}\;\; \text{or}\;\; a=.76125127398793390138765377686933423...$$
$a$ is increasing as a increases. The other formula decreases as $a$ increases
$$a = 2^{\frac{2}7}/3^{\frac{3}7}$$ is when both sides are equal. Any change in a in either direction will cause one of them (and therefore the minimum) to decrease.
Your work up to that point was correct.
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