Question

Find stationary points of following derived function...

Answer 1

\[g'(x)=1/2(4\cos^2 x+ 4\cos x-3) \exp ^{1/2x +cosx}\]

Answer 2

You want to find the stationary point of g'(x)?

Answer 3

Yes

Answer 4

It is all differentiated

Answer 5

Could you find g"(x)?

Answer 6

I think I need to find the value for g'(x) =0 but not sure which part to use if not all of the function

Answer 7

Exponential can't be zero, equate the other terms to zero and find values of x

Answer 8

(2cosx-1)(2cosx+3) =0
so x = 1 and x= -3
Are these the stationary points

Answer 9

uh huh
\[\cos x= \frac{1}2 \text {and} \cos x= \frac{2}{3}\]
Now find x ?

Answer 10

This is where I am really stuck. How would I do that?

Answer 11

brb

Answer 12

I'm back:)
Could you tell me the value of x for which cos x =1/2 ?

Answer 13

60

Answer 14

Yeah it's 60 or \(\frac{\pi}3\) but there are several values of x for which cos x=1/2.
Could you tell me?

Answer 15

@hmidante ?

Answer 16

2pi/3
4pi/3
7pi/4

Answer 17

\[\cos \frac{\pi}{3}=\frac 12\]
cos is negative in the second and third quadrants
it's positive in first and fourth
so in fourth it's 1/2 at \(300\ degrees\ or\ 2\pi-\frac{\pi}{3} or \frac{5\pi}3\)
Do you get this?

Answer 18

Yes

Answer 19

so for cos x=1/2
we have
\[x=2n\pi \pm \frac{\pi}{3}\ n=0, 1, 2, 3..\]

Answer 20

Do you get this?

Answer 21

Not really? Could explain a bit. This if the final part that I really need to understand to find out what x really means in terms of this function and graph.

Answer 22

We found that x= pi/3 and x= 5pi/3 for cos x=1/2
did you get this part?

Answer 23

yes

Answer 24

and you know that cos x repeats after 2pi?

Answer 25

Yes

Answer 26

so if we add or subtract 2pi to the x values, cos x will still be 1/2

Answer 27

Ok.

Answer 28

That's why the general solution is
\[x= 2n\pi \pm \pi/3\]
put n=0 with + sign, you'll get pi/3 , n=1 and - sign gives 5pi/3 and so we can obtain all solutions

Answer 29

and cos x = 2/3 for example?

Answer 30

Yeah for that, we don't know what's x value in degrees or radians
so we can leave the solution as
\[x=2n\pi\pm \cos^{-1} (\frac 23)\]

Answer 31

the interval is 0 to 2pi

Answer 32

I think in radians then?

Answer 33

In the interval 0 to 2pi
we'll have only 4 solutions.
Could you tell me?

Answer 34

0
1/2
2/3
0

Answer 35

?

Answer 36

for 1/2 x= pi/3 and 5pi/3
for 2/3
\[x=\cos^{-1} \frac 23\ and\ 2\pi-\cos^{-1} \frac 23\]

Answer 37

Do you get this?

Answer 38

Think so. So the x values would be 48 and -41

Answer 39

This would be on the x line?

Answer 40

48 and (360-48)
in degrees

Answer 41

Ok. and hopw would i translate that into the x line?

Answer 42

x=60, 300, 48, 312
in degrees
to convert too radians divide the degrees by 57