Finding the maximum of a function on a specific interval

Question

I have a problem at it is as follows. I've to find the maxium value of the following function between to time points, namely $t=0$ and $t=\frac{2}{50}$. The function is the following:

$$8-\exp(-\frac{2500t}{3})+(4\exp(-\frac{125(20t-1)}{3})-8)\cdot\theta(t-\frac{1}{20})$$

Where $\theta$ is the HeavisideTheta function.

I did the following analysis:

$$\frac{d}{dt}\left\{8-\exp(-\frac{2500t}{3})+(4\exp(-\frac{125(20t-1)}{3})-8)\cdot\theta(t-\frac{1}{20})\right\}=0\rightarrow$$ $$t=$$

But that makes it realy hard. So is there a smart way to solve this?

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This problem is coming from a physics problem.

Answer 1

$\tfrac{2}{50}<\tfrac{1}{20}$, so in $[0,2/50]$ your function is just $$ 8-\exp\left(-\frac{2500t}3\right), $$ which is obviously increasing, so the maximum is at the right endpoint and its value is $$ 8-\exp\left(-\frac{100}3\right). $$

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Also, notice that your original function is discontinuous. I don't know if this is ok or by mistake. Probably by mistake...