Find the when the maximum happens of this function.

Question

Myself and some coworkers have attempted to differentiate the function, solve for the min.

We have gotten that a maximum happens at x=ln(b/a)/(b-a), but the online platform is saying it’s wrong and there needs to be c’s involved??

Can someone attempt and explain why I’m wrong/ online platform is wrong?

Answer 1

You are correct. The $c$ scales the overall concentration, but does not affect the time at which a maximum occurs. You can think of $c$ as merely setting the units you measure the concentration in.

Answer 2

The constant in front is irrelevant so let's differentiate $(e^{-at}-e^{-bt})'=-ae^{-at}+be^{-bt}=0 \to e^{-bt+at}=\frac{a}{b}$ or $t(a-b)=\ln a - \ln b$, $t=\frac{\ln a - \ln b}{a-b}$. Your answer is the same.