Question

Consider the equation below. (if an answer does not exist, enter dne.) f(x) = 5 sin(x) + 5 cos(x), 0 ? x ? 2? (a) find the interval on which f is decreasing. (enter your answer using interval notation.)

Answer 1

Answer: $\bold{[\dfrac{\pi}{4},\dfrac{5\pi}{4}]}$

Step-by-step explanation:

Step 1: Create a table

x    |   5sinx + 5cosx = y

0    |       0    +     5     =   5

$\frac{\pi}{2}$   |        5    +     0    =   5

π    |        0    +    -5    =  -5

$\frac{3\pi}{2}$    |       -5    +     0    =  -5

2π |        0    +      5    =   5

Notice that y = 5 at 0 and $\frac{\pi}{2}$ , so there will be a vertex at their midpoint.  Similarly at y = -5.

Midpoint of 0 and $\frac{\pi}{2}$  is $\dfrac{\pi}{4}$ . Midpoint of π and $\frac{3\pi}{2}$ is $\dfrac{5\pi}{4}$

(graph is attached to confirm interval)