Question

Show that r(x) = 2x-cos^2(x)+sqrt(2) has exactly one zero on (-inf,inf)

I have that the first derivative is: 2+sin2x > 0 on (-inf,inf) Now what? I think I need to use the Intermediate Value Theorem to show that there's at least one zero on the interval, and then Rolle's Theorem to show there's exactly one zero on the interval, but I'm not sure where to start.

Answer 1

no you don't need any such thing.

Answer 2

function is continuous for sure, and since the derivative is positive that tells you it is always increasing

Answer 3

so find some value that gives a negative number, one that gives a positive number, and since it is continuous it must be zero somewhere. also because it is increasing it cannot be zero more than once