Find the absolute extreme values of the function on the interval.
-$$f ( x ) = \tan x , - \frac { \pi } { 4 } \leq x \leq \frac { \pi } { 6 }$$
A) absolute maximum is $\frac { \sqrt { 3 } } { 3 }$ at $x = \frac { \pi } { 6 }$; absolute minimum is $- 1$ at $x = - \frac { \pi } { 4 }$
B) absolute maximum is $\frac { \sqrt { 3 } } { 3 }$ at $x = \frac { 2 \pi } { 18 }$; absolute minimum is $- 1$ at $x = - \frac { \pi } { 8 }$
C) absolute maximum is $- 1$ at $x = \frac { \pi } { 6 }$; absolute minimum is $\frac { \sqrt { 3 } } { 3 }$ at $x = - \frac { \pi } { 4 }$
D) absolute maximum is $\frac { \sqrt { 3 } } { 3 }$ at $x = \frac { \pi } { 6 }$ and $- \frac { \pi } { 4 }$; absolute minimum does not exist

Question

Quizplus · Accepted Answer

Firstly, note that $	an x$ is undefined at $x = \frac{\pi}{2}, \frac{3\pi}{2},\frac{5\pi}{2},\dots$, but all such values are outside the given interval. So, we can focus on checking the critical points and endpoints of the interval.

The function $	an x$ is continuous on the interval $-\frac{\pi}{4} \leq x \leq \frac{\pi}{6}$, so by the Extreme Value Theorem, both an absolute maximum and an absolute minimum exist on this interval.

To find these extreme values, we first find the critical points in the interval. These are the points where the derivative of the function is zero or undefined. We have:

$$\frac{d}{dx}(	an x) = \sec^2 x$$

This is 0 at no points in the interval, but it is undefined at $x = \frac{\pi}{2}, \frac{3\pi}{2},\frac{5\pi}{2},\dots$. Since these points are outside the interval, there are no critical points in the interval.

Next, we check the endpoints of the interval. We have:

$$f \left(-\frac{\pi}{4}ight) = 	an \left(-\frac{\pi}{4}ight) = -1$$
$$f \left(\frac{\pi}{6}ight) = 	an \left(\frac{\pi}{6}ight) = \frac{\sqrt{3}}{3}$$

Thus, the absolute minimum of $f$ is $-1$ at $x = -\frac{\pi}{4}$ and the absolute maximum of $f$ is $\frac{\sqrt{3}}{3}$ at $x = \frac{\pi}{6}$. So, the answer is (A).

Find the Absolute Extreme Values of the Function on the Interval