How do you use the intermediate value theorem to verify that there is a zero in the interval [0,1] for g(t)=2cost-3t?

1 Answer
Aug 22, 2016

There is no zero for g(t)=2cost-3t in the range [0,1].

Explanation:

It is observed that both cost and 3t are continuous in the range [0,1], and hence g(t)=2cost-3t is also continuous over the range [0,1].

Now g(0)=2cos0-3×0=2-3=-1 and g(1)=2cos1-3=2×0.5403-3=-1.9194.

As g(t) is continuous but does not change sign between [0,1], we do not have a zero in the range [0,1]. In fact as the derivative of g(t), g'(t)=-2sint-3 is negative in the range g(t) is continuously decreasing in the range and never reaches the value 0.