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- Thread starter mt91
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- Jan 30, 2018

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Problem (6) asks about the "yield" which is defined as Y= Eu*(E) where u* is a "steady state solution". Since the steady state solutions are u*(E)= 0 and u*(E)= sqrt(1- E), either Y= E(0)= 0 or Y= E sqrt(1- E)= sqr(E^2- E^3). The first is identically equal to 0 so cannot be maximized. To find the maximum of the second, set the derivative equal to 0.

Y= sqrt(E^2- E^3)= (E^2- E^3)^(1/2). Y'= (1/2)(E^2- E^3)^(-1/2)(2E- 3E^2)= 0. That is equivalent to 3E^2- 2E= E(3E- 2)= 0 so E= 0 or E= 2/3. Again, E= 0 cannot give a maximum (it gives a minimum) so E*= 2/3.

Problem (6) asks about the "yield" which is defined as $Y= Eu^*(E)$ where u* is a "steady state solution". Since the steady state solutions are $u^*(E)= 0$ and $u^*(E)= \sqrt(1- E)$, either $Y= E(0)= 0$ or $Y= E \sqrt(1- E)= \sqrt(E^2- E^3)$. The first is identically equal to 0 so cannot be maximized. To find the maximum of the second, set the derivative equal to 0.

$Y= \sqrt(E^2- E^3)= (E^2- E^3)^{1/2}$. $Y'= (1/2)(E^2- E^3)^(-1/2)(2E- 3E^2)= 0$. That is equivalent to $3E^2- 2E= E(3E- 2)= 0$ so $E= 0$ or $E= 2/3$. Again, $E= 0$ cannot give a maximum (it gives a minimum) so $E^*= 2/3$.