rolles theorem can be applied to f(x) = tan(x) over the interval -π/2, 3π/2 to show there is some value -π/2…

Explanation:

Step1: Recall Rolle's Theorem conditions

Rolle's Theorem requires a function $y = f(x)$ to be continuous on $[a,b]$, differentiable on $(a,b)$ and $f(a)=f(b)$.

Step2: Analyze continuity of $f(x)=\tan(x)$

The function $y = \tan(x)$ has vertical asymptotes at $x=-\frac{\pi}{2}$ and $x = \frac{\pi}{2}$. It is not continuous on the interval $\left[-\frac{\pi}{2},\frac{3\pi}{2}\right]$ since it has infinite - discontinuities at the endpoints (in the extended sense for this interval).

Step3: Conclusion

Since $f(x)=\tan(x)$ is not continuous on $\left[-\frac{\pi}{2},\frac{3\pi}{2}\right]$, Rolle's Theorem cannot be applied.

Answer:

False