I have a function given by: $$2\cos(100t)+18\cos(2000\pi t).$$ Its maximum value will be at $t=0$ and will be $2+18=20.$
But how one should find its minimum value. Its minimum value can't be $-20$ as: $$100t=(2n+1)\pi$$ $$2000\pi t=(2m+1)\pi$$ and equating the two terms we have time $t$ at which we get $-20$ to be: $$\frac{(2n+1)\pi}{100}=\frac{(2m+1)}{2000}$$ which does not seem to be possible as one side is rational and another is irrational, so there is no value of $n$ and $m$ to satisfy these equations. How should I then find its minimum value? Can differentiation help? because there also I will get sinusoidal terms.
There is no minimum value, but you can get as close to $-20$ as you want. Because the periods are not rationally related, there is at most one point where the peaks line up. Here it is $t=0$. You can find values of $t$ that make it very close to $-20$ but not that hit $-20$