Question

Stuck on a problem finding where a function is increasing or decreasing

f(x)=5sin^2x on interval [-pi,pi]

I know the derivative f'(x)=10sinxcosx
and then set it = 0

10sinxcosx=0

that is where im stuck because you cant divide or move anything and im not sure how to approach the problem

Answer 1

to find where the function is increasing, you solve f ' (x) > 0

Answer 2

f ' (x) = 0 gives you just the x value where f has a slope of zero

Answer 3

you want to find x values where f ' (x) is positive (so f is increasing)
and x values where f ' (x) is negative ( so f is decreasing)

Answer 4

so
10cosxsinx > 0
and then plug in variables until i find one that gives a value greater than 0 and a value less than 0?

Answer 5

set 10*cos(x)*sin(x) equal to zero and solve for x. Once you get the roots of the equation, you can figure out what values to test (so you can see which parts are above the x axis)

Answer 6

thats where im not sure how to approach the problem. i set it = 0
and then the only thing i can do would be to divide and if i do that i would just get 0=0

Answer 7

10*cos(x)*sin(x) = 0

cos(x) = 0 or sin(x) = 0

solve each piece for x

Answer 8

use the unit circle

Answer 9

keep your interval of [-pi, pi] in mind

Answer 10

OHHH
ok
i didnt even think to break it apart like that, give me a minute to work it

Answer 11

So for sinx it would be x=0
and for cosx it would be x = -pi/2 and pi/2
i googled a picture of a unit circle and i am pretty sure that is correct

Answer 12

there's one more solution to sin(x) = 0

Answer 13

positive pi?

Answer 14

-pi as well, sorry 2 other solutions

Answer 15

so you have these 5 roots in [-pi, pi]

-pi, -pi/2, 0, pi/2, pi

Answer 16

so there is two for cosx which are -pi/2 and pi/2
and three for sinx are -pi 0 pi

Answer 17

yes

Answer 18

those are your boundary points so to speak

Answer 19

anything in between needs to be tested (one value per region) to see if 10*cos(x)*sin(x) > 0 is true or not

Answer 20

and thats where i would plug in values from inbetween those points to determine if it is increasing or decreasing at certain intervals

Answer 21

yes

Answer 22

I recommend a sign chart

Answer 23

this page explains it a bit more with actual visual graphs to show

http://www.purplemath.com/modules/ineqsolv3.htm

Answer 24

if i could give more than 1 best response i would thanks for the pointers. im going to have to do a little review on the unit circle. but thanks for the help

Answer 25

np