Show that the functions in this interval have exactly one zero in the given interval,?

Question

Show that the functions in this interval have exactly one zero in the given interval,

~ This is inside the rolle's theorem, Mean Value Theorem section in calculus.

F(x)= x + (sin(x/3))^2 -8 on the interval (-infinity, infinity)

Answer 1

Hmm. Without doing the whole problem, I know where I'd start. sin(anything) is between -1 and 1. So sin^2 is between 0 and 1. I.e., x-8 <= F(x) <= x-7 everywhere.

So the only possible values of x for which there could be zeros lie in the interval [7,8]. Solve the problem there, and you'll have solved it overall.