Expert: Paul Klarreich - 8/15/2009

Question
I have to find the stationary points of g(x)=cosxexp(2/3sinx) (0<x<2pi) and then use the first derivative test to classify them . I can do this with functions that do not contain the trigonometric identities , but am really struggling here. Thank you in advance.

Answer
Questioner: Jo
Country: United Kingdom
Category: Advanced Math
Private: No
Subject: Advanced Maths
Question: I have to find the stationary points of g(x)=cosxexp(2/3sinx) (0<x<2pi) and then use the first derivative test to classify them . I can do this with functions that do not contain the trigonometric identities , but am really struggling here. Thank you in advance.
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g' =  (cos x)(2/3 cos x exp(2/3 sin x) + (exp(2/3 sin x))(- sin x)

g' =  (cos x)(2/3 cos x exp(2/3 sin x) + (exp(2/3 sin x))(- sin x)

g' = exp(2/3 sin x)[ 2/3 cos^2(x) - sin x]

Set that equal to zero and solve.  
>> From here on, ignore the exp(..) -- that is always positive

Use  cos^2(x) = 1 - sin^2(x):

2/3(1 - sin^2(x)) - sin x = 0

2/3(1 - s^2) - s = 0

2(1 - s^2) - 3s = 0

2 - 2s^2 - 3s = 0

2s^2 + 3s - 2 = 0

(2s - 1)(s + 2) = 0

>> From here on, ignore the  sin x + 2; that is always positive.

sin x = 1/2

x = pi/6 or  x = 5pi/6    << You will have to review your trig for this.

Now you use your FIRST DERIV test:

If x < pi/6, then (2s - 1)(s + 2) is (pos, neg)??
If x > pi/6, then (2s - 1)(s + 2) is (pos, neg)??

If x < 5pi/6, then (2s - 1)(s + 2) is (pos, neg)??
If x > 5pi/6, then (2s - 1)(s + 2) is (pos, neg)??

If you decide that the answers are like this:

If x < pi/6, then (2s - 1)(s + 2) is pos.
If x > pi/6, then (2s - 1)(s + 2) is neg.

then you say MAX.

If you decide that the answers are like this:

If x < pi/6, then (2s - 1)(s + 2) is neg.
If x > pi/6, then (2s - 1)(s + 2) is pos.

then you say MIN.

Analyze like this:

If  x < pi/6, then  sin x < 1/2,
If  x < pi/6, then  2 sin x < 1,
If  x < pi/6, then  2 sin x - 1 < 0, so NEG.

etc.

I'll leave the rest to you.