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

Answer:

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#.