Question 996014: Let g(x) = 8x + sin 8x for 0 ≤ x < 2π.
The function g will have ____critical points in [0,2pi)
List in increasing order the first two critical points in [0, 2π). Enter DNE for any empty answer blank.
x =
x =
This problem is driving me bonkers! I guess I am confused with how the domain works with this . I got to cos(8x) =-1 part and then did a U-sub to find out what 8x is, I know cos(u)=-1 then u must be pi. I got pi/8 but I am confused beyond belief on how to find the increasing order of this or what it means. Like how it gets to be 3pi/8, 5pi/8 etc.
Thank you for any and all help!
Answer by jim_thompson5910(35256)   (Show Source):
You can put this solution on YOUR website! g(x) = 8x + sin(8x)
g ' (x) = 8 + 8*cos(8x)
0 = 8 + 8*cos(8x)
8*cos(8x) = -8
cos(8x) = -8/8
cos(8x) = -1
8x = arccos(-1)
8x = pi + 2pi*n ... where n is an integer
x = (pi + 2pi*n)/8
If we made a table with n as the input, x as the output, then...
| n | x (exact) | x (approximate) |
|---|
| -1 | -pi/8 | -0.39270 | | 0 | pi/8 | 0.39270 | | 1 | 3pi/8 | 1.17810 | | 2 | 5pi/8 | 1.96350 | | 3 | 7pi/8 | 2.74889 | | 4 | 9pi/8 | 3.53429 | | 5 | 11pi/8 | 4.31969 | | 6 | 13pi/8 | 5.10509 | | 7 | 15pi/8 | 5.89049 | | 8 | 17pi/8 | 6.67588 |
When n = -1, the value of x is -0.39270 which is outside the interval [0,2pi). The same applies to when n = 8 (x = 6.67588). So we only focus on when n = 0 all the way through n = 7.
We can see that there are 8 critical points (7-0+1 = 8) since there are 8 x values in the interval [0,2pi).
The first two critical points are x = pi/8 and x = 3pi/8
pi/8 = 0.39270 approx
3pi/8 = 1.17810 approx
This corresponds to n = 0 and n = 1 respectively.
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Answer:
The function g will have 8 critical points in [0,2pi)
List in increasing order the first two critical points in [0, 2pi). Enter DNE for any empty answer blank.
x = pi/8
x = 3pi/8
If the book is seeking approximate x values (instead of exact), then the two answers for x are
x = 0.39270
x = 1.17810
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