Answer
Peak value: $2\sqrt 2$ amps.
Work Step by Step
Step 1. Given the function $i(t)=2cos(t)+2sin(t)$, we can find its derivative as $i'(t)=2cos(t)-2sin(t)$.
Step 2. The critical points can be found when $i'=0$ or undefined which happens when $tan(t)=1, t=k\pi+\frac{\pi}{4}$ where $k$ is an integer.
Step 3. We only need to find the peak value which can be obtained with a single critical point. At $t=\frac{\pi}{4}$, $i(\frac{\pi}{4})=2cos(\frac{\pi}{4})+2sin(\frac{\pi}{4})=2\sqrt 2$ amps and we can identify this as a maximum (use local test points as necessary) as shown in the figure.
