How do I show that the function $e^{−4x}+2\cos(20x)=0$ has a unique root within the interval $[0, \frac{\pi}{20}]$?
I tried using the Mean Value Theorem to show that, if supposed there exist two roots within the interval then the two roots are the same. (Proof by contradiction) But it turned out bad and too complicated. (I am not sure if I did the right thing by using the MVT here)
I also tried to find the $f'(x)$ and then finding the turning points to prove that the function is monotonically increasing or decreasing, but same thing. It is too hard to find any turning point when $f'(x)=0$ for this specific function.
Any other tips on how I would be able to go through this?
Any help is appreciated
Best Answer
Note that the derivative of the function is negative on the interval $[0; \frac{\pi}{20}]$.
Since the function has opposite signs at the extremes of the interval (it goes from positive to negative) and is monotonic, it will have a single zero.