The survival of a fish egg through its critical period is a function of its mass, $x$. The larger the egg, the more nutrients are present and the more likely it is to hatch successfully. This survivorship is often modeled by a function of the form
$s(x)=1 − cx^{-b}$ , $x ≥ c$, where c and b are positive parameters.
If $G$ is the total gonadal mass of the female, the number of eggs laid by the female is $G/x$, and the number of eggs that survive through the critical period is $E(x) = G/ x (1 − cx^{−b}) $. Show that the egg size that optimizes the female’s number of eggs is $x^∗ = (c + bc) ^{1/b}$
I am stuck on this problem, it would seem that it is a simple optimization problem, where you take the derivative set it to zero and solve, but is it really that simple? The derivative I get with respect to $E(x)$ is $bcgx^{-b-2} - g * ((1-cx^{-b})/(x^2)) $ Is this correct?
Thank you
My CAS (Sage) indeed finds the same derivative as yours, and you can then factor it as ${\left(b c + c - x^{b}\right)} G x^{-b - 2}$, from which the final answer can be obtained.
I assume that 'the egg size that optimizes the female’s number of eggs' actually means 'the egg size that maximizes the number of eggs that survive'. In order to check that the solution corresponds to a maximum, you probably want to check the signs of the second derivative.