MHB What is the connection between roots of f and g using Rolle's Theorem?

[Tomp]

[h=1][/h]I'm doing a question and I am getting stuck and need help

Question:
Consider the continuous functions f(x) = 1 - e^(x)*sin(x) and g(x) = 1 + e^(x)*cos(x). Using Rolle's Theorem, prove that between any two roots of f there exists at least one root of g.

Hint
Remember that, a root of f is a point x in the domain of f such that f(x) = 0.

Can someone provide a natural language proof of this?

 

[MarkFL]

Let a and b be two successive roots of f. Observe that by Rolle's theorem, we must have some c in (a,b) such that f'(c)=0. This means we must have:

$\displaystyle -e^c\cos(c)-e^c\sin(c)=0$

$\displaystyle -e^c\sin(c)=e^c\cos(c)$

$\displaystyle 1-e^c\sin(c)=1+e^c\cos(c)$

$\displaystyle f(c)=g(c)$

So, we find that f and g meet at the turning points of f. This means g must have at least one root between two successive turning points of f.