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Differentiation Question

  1. Jan 15, 2010 #1
    1. The problem statement, all variables and given/known data
    Find a function of the form [tex] f(x) = a + b \cos cx [/tex] that is tangent to the line [tex]y = 1[/tex] at the point [tex](0,1)[/tex], and tangent to the line [tex]y = x + 3/2 - \pi /4[/tex] at the point [tex](\pi /4 , 3/2)[/tex].

    2. Relevant equations

    3. The attempt at a solution
    [tex]f(0) = a + b = 1[/tex], so [tex]a = 1 - b[/tex].

    This is as far as I can get though.

    [tex]f'(0) = -bc \sin cx = 0[/tex]

    for any a, b, and c, and

    [tex]f(\pi /4) = (1 - b) + b \cos [(\pi /4)c] = 3/2[/tex]


    [tex]f'(\pi /4) = -bc \sin [(\pi /4)c] = 1[/tex]

    don't really seem to help me.

    What am I missing?
  2. jcsd
  3. Jan 16, 2010 #2
    Well you can combine the last two equations to get
    [tex]b\left[\cos\left(\frac{c\pi}{4}\right)-c\sin\left(\frac{c\pi}{4}\right)-1\right] = \frac{3}{2}.[/tex]
    Presumably this will give you infinitely many solutions. For instance, c = 2 works.
  4. Jan 16, 2010 #3
    the function has a maximum at x = 0, because ymax = a+b
    if you do the second derivative test you will find -bc^2 < 0
    so b>0

    i am not able to tell more than this from the given data
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