step 3\nif ( 0<x<\frac{pi}{4} ), then ( f^{prime}(x) ) is positive, and ( f(x) ) is increasing.\nstep 4\nif…

step 3\nif ( 0<x<\frac{pi}{4} ), then ( f^{prime}(x) ) is positive, and ( f(x) ) is increasing.\nstep 4\nif ( \frac{pi}{4}<x<\frac{5 pi}{4} ), then ( f^{prime}(x) ) is negative, and ( f(x) ) is decreasing.\nstep 5\nif ( \frac{5 pi}{4}<x<2 pi ), then ( f^{prime}(x) ) is positive, and ( f(x) ) is increasing.\nstep 6\ntherefore, the interval on which ( f ) is increasing is the following. (enter your answer using interval notation.)\n\nthe interval on which ( f ) is decreasing is the following. (enter your answer using interval notation.)\n

step 3\nif ( 0<x<\frac{pi}{4} ), then ( f^{prime}(x) ) is positive, and ( f(x) ) is increasing.\nstep 4\nif ( \frac{pi}{4}<x<\frac{5 pi}{4} ), then ( f^{prime}(x) ) is negative, and ( f(x) ) is decreasing.\nstep 5\nif ( \frac{5 pi}{4}<x<2 pi ), then ( f^{prime}(x) ) is positive, and ( f(x) ) is increasing.\nstep 6\ntherefore, the interval on which ( f ) is increasing is the following. (enter your answer using interval notation.)\n\nthe interval on which ( f ) is decreasing is the following. (enter your answer using interval notation.)\n

Answer

Explanation:

Step1: Analyze increasing intervals

From Step 3, when (0 < x<\frac{\pi}{4}), (f(x)) is increasing. From Step 5, when (\frac{5\pi}{4}<x < 2\pi), (f(x)) is increasing. In interval notation, we combine these two intervals.

Step2: Analyze decreasing intervals

From Step 4, when (\frac{\pi}{4}<x<\frac{5\pi}{4}), (f(x)) is decreasing.

Answer:

The interval on which (f) is increasing: (\left(0,\frac{\pi}{4}\right)\cup\left(\frac{5\pi}{4},2\pi\right)) The interval on which (f) is decreasing: (\left(\frac{\pi}{4},\frac{5\pi}{4}\right))