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Application of derivatives

  1. Dec 21, 2015 #1
    1. The problem statement, all variables and given/known data
    the original function is ##−6 x^3−3x−2 cosx##

    ##f′(x)=−2x^2−3+2sin(x)##
    ##−2x^2 ≤ 0## for all x
    and ##−3+2 sin(x) ≤ −3+2 = −1##, for all x
    ⇒ f′(x) ≤ −1 < 0 for all x

    3. The attempt at a solution

    this problem is part of a larger problem which says
    there is a cubic function which can have at least one real roots
    than we prove (like the above) that the derivative is negative and therefore the function is strictly decreasing (so it cannot intersect the x-axis again to have another root coz it cannot increase)
    therefore it has exactly one real root (not three)

    my problem is how they proved that ##-3+2 sin(x)## is -1
    how is the sin(x) value +1
    does it not oscillate b/w -1 and 1?
     
  2. jcsd
  3. Dec 21, 2015 #2

    mfb

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    The derivative is not correct, and the original function is not a cubic function due to the cosine.

    It does oscillate, but a smaller sine value makes the inequality even stronger.
    That's why the two parts are connected via ##\leq## and not =.
     
  4. Dec 21, 2015 #3
    yeah the derivative is wrong x^2 has a coefficient -18
    and the inequality is correct (I was not paying attention)
    does cosine(and trig fns) also change the degree? I thought that only the highest power of x indicated the degree
     
  5. Dec 21, 2015 #4

    mfb

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    Cosine and sine are not powers of their argument, the function is not a polynomial function of any degree.
     
  6. Dec 21, 2015 #5
    So,How do you think about the max no of roots for the function?
     
  7. Dec 21, 2015 #6

    mfb

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    It has at most one root, as shown in the first post. The proof via the derivative works for all functions with a derivative, they don't have to be polynomial functions.
     
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