determine whether rolles theorem applies to the given function on the given interval. if so, find the…

determine whether rolles theorem applies to the given function on the given interval. if so, find the point(s) that are guaranteed to exist by rolles theorem.\n\n$f(x)=3\\tan(2x);\\left\\frac{\\pi}{4},\\frac{3\\pi}{4}\\right$\n\nsince $f$ is continuous on the closed interval $\\left\\frac{\\pi}{4},\\frac{3\\pi}{4}\\right$, differentiable on the open interval $\\left(\\frac{\\pi}{4},\\frac{3\\pi}{4}\\right)$, and $f\\left(\\frac{\\pi}{4}\\right)\\square\\nabla f\\left(\\frac{3\\pi}{4}\\right)$, rolles theorem $\\square\\nabla$ be applied to $f$ on the interval $\\left\\frac{\\pi}{4},\\frac{3\\pi}{4}\\right$.

determine whether rolles theorem applies to the given function on the given interval. if so, find the point(s) that are guaranteed to exist by rolles theorem.\n\n$f(x)=3\\tan(2x);\\left\\frac{\\pi}{4},\\frac{3\\pi}{4}\\right$\n\nsince $f$ is continuous on the closed interval $\\left\\frac{\\pi}{4},\\frac{3\\pi}{4}\\right$, differentiable on the open interval $\\left(\\frac{\\pi}{4},\\frac{3\\pi}{4}\\right)$, and $f\\left(\\frac{\\pi}{4}\\right)\\square\\nabla f\\left(\\frac{3\\pi}{4}\\right)$, rolles theorem $\\square\\nabla$ be applied to $f$ on the interval $\\left\\frac{\\pi}{4},\\frac{3\\pi}{4}\\right$.

Answer

Explanation:

Step1: Check continuity

The function (y = \tan t) has vertical asymptotes at (t=\frac{\pi}{2}+k\pi,k\in\mathbb{Z}). For (f(x)=3\tan(2x)), when (x = \frac{\pi}{4}), (2x=\frac{\pi}{2}); when (x=\frac{3\pi}{4}), (2x=\frac{3\pi}{2}). The function (f(x)) is not continuous on the closed interval (\left[\frac{\pi}{4},\frac{3\pi}{4}\right]) because of the vertical asymptotes at (x = \frac{\pi}{4}) and (x=\frac{3\pi}{4}).

Answer:

Since (f) is not continuous on the closed interval (\left[\frac{\pi}{4},\frac{3\pi}{4}\right]), Rolle's Theorem cannot be applied to (f) on the interval (\left[\frac{\pi}{4},\frac{3\pi}{4}\right]).