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anonymous

  • one year ago

Consider the equation below. f(x) = 3 cos^2x − 6 sin x, 0 ≤ x ≤ 2π (a) Find the interval on which f is increasing. (Enter your answer in interval notation.) b)Find the interval on which f is decreasing. (Enter your answer in interval notation.) c)(b) Find the local minimum and maximum values of f. d)(c) Find the inflection points. e)Find the interval on which f is concave up. (Enter your answer in interval notation.) f)Find the interval on which f is concave down. (Enter your answer in interval notation.) Please help me work this out!

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  1. anonymous
    • one year ago
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    \[f(x)=3\cos^2(x)-6\sin(x)\] right?

  2. anonymous
    • one year ago
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    yes

  3. anonymous
    • one year ago
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    did you take the derivative as a first step?

  4. anonymous
    • one year ago
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    and if so, what did you get?

  5. anonymous
    • one year ago
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    -6cos(x)sin(x)-6cos(x)

  6. anonymous
    • one year ago
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    that looks good i guess the next step is to factor

  7. anonymous
    • one year ago
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    you got that?

  8. anonymous
    • one year ago
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    once you factor out the \(-6\cos(x)\) it is going to be real easy to see where the derivative is positive and negative

  9. anonymous
    • one year ago
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    -6cos(x)(sin(x)+1)?

  10. anonymous
    • one year ago
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    yeah and forturnately for you, since \(\sin(x)\geq -1\) always, you know \(\sin(x)+1\geq 0\) so you can ignore that part when you check tor the sign of the derivative

  11. anonymous
    • one year ago
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    in other words, the sign of the derivative is completely dependent on the sign of \(-6\cos(x)\)

  12. anonymous
    • one year ago
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    you got that?

  13. anonymous
    • one year ago
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    no, im confused as to what my answer would be

  14. anonymous
    • one year ago
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    ok lets go slow

  15. anonymous
    • one year ago
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    it is clear that in order to find the interval over which the function is increasing (decreasing) your only job is to find the interval over which the derivative is positive (negative) yes?

  16. anonymous
    • one year ago
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    yes

  17. anonymous
    • one year ago
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    ok and your derivative isi \[-6\cos(x)\left(\sin(x)+1\right)\]

  18. anonymous
    • one year ago
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    so the question is, over what interval is the derivative positive

  19. anonymous
    • one year ago
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    so i would have to graph it to see it clearly right?

  20. anonymous
    • one year ago
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    you have two factors \[-6\cos(x)\] and \[\sin(x)+1\] no you do not have to graph it

  21. anonymous
    • one year ago
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    is it clear that \[\sin(x)+1\geq 0\] for any value of \(x\)?

  22. anonymous
    • one year ago
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    yes

  23. anonymous
    • one year ago
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    that means when you are trying to find if the derivative is positive or negative, you can ignore that factor entirely (because it is never negative) and concentrate only on \(-6\cos(x)\)

  24. anonymous
    • one year ago
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    do you know over what intervals between \(0\) and \(2\pi\) that cosine is positive?

  25. anonymous
    • one year ago
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    i cant remember

  26. anonymous
    • one year ago
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    then i guess i have to tell you

  27. anonymous
    • one year ago
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    it is positive on \((0,\frac{\pi}{2})\) that is on the right sides of the unit circle

  28. anonymous
    • one year ago
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    negative on \((\frac{\pi}{2},\frac{3\pi}{2})\) that is the left side

  29. anonymous
    • one year ago
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    and positive again on \((\frac{3\pi}{2},2\pi)\)

  30. anonymous
    • one year ago
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    since your cosine has a \(-6\) in front of it, it will be negative where cosine is positive, and vice versa

  31. anonymous
    • one year ago
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    okay so for a) it would be pi/2,3pi/2 ?

  32. anonymous
    • one year ago
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    increasing on that interval, yes

  33. anonymous
    • one year ago
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    and of course decreasing on the remaining intervals

  34. anonymous
    • one year ago
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    to find the max and min i would plug in to the equation pi/2 and then 3pi/2?

  35. anonymous
    • one year ago
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    yes

  36. anonymous
    • one year ago
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    in to the original function of course, not the derivative

  37. anonymous
    • one year ago
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    got 6 and -6

  38. anonymous
    • one year ago
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    looks good

  39. anonymous
    • one year ago
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    thank you so much!

  40. anonymous
    • one year ago
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    yw

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