Question

Use the intermediate value theorem to determine if
h[theta]=1- 3 tan theta
has a zero in the interval [0,pi/4].
Use form IVT to answer

Answer 1

If \(h(\theta)\) is continuous over than interval, you should be able to check to signs on the endpoints, then determine if \(h\) crosses the \(\theta\)-axis.

Answer 2

So yes, it is continuous, because the domain of the tangent function is \(\left(-\dfrac{\pi}{2},\dfrac{\pi}{2}\right)\).

Evaluate the endpoints:
\[h(0)=1-3\tan0=1-0=1\\
h\left(\frac{\pi}{4}\right)=1-3\tan\frac{\pi}{4}=1-3=-2\]
Since \(h(\theta)\) changes from positive to negative values over the interval, and since it's continuous over this interval, there must be some value \(c\) such that \(h(c)=0\).