Question

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`?

Answer 1

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`.