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Inflection Points question

  1. Nov 12, 2009 #1
    1. The problem statement, all variables and given/known data
    f(x)=-8x4-5x3 +3 for concavity and inflection points.


    2. Relevant equations
    f(x)=-8x4-5x3 +3
    f'(x)=-32x3-15x2
    f''(x)=-96x2-30x

    3. The attempt at a solution
    can someone explain to me what exactly Inflection Points are?
    is it like critical points for the first derivative but those are for the second one?
    i mean, making f''(x) = 0 and get 2 numbers and those are the Inflection Points?
     
  2. jcsd
  3. Nov 12, 2009 #2
    Indeed. Inflection points are where the function changes from being concave-up to concave-down or vice versa. This is the same as the local extrema of the derivative, or the roots of the second derivative.
     
  4. Nov 12, 2009 #3
    so i would do this
    f''(x)=-96x2-30x
    f''(x)=-3x(32x-10)

    x=0, 10/32

    and those are my answers? well i tried and it doesnt work.. :(
     
  5. Nov 12, 2009 #4

    Mark44

    Staff: Mentor

    f''(x) = -96x2 - 30x = -6x(16x + 5)
    So f''(x) = 0 for x = 0 and x = -5/16
     
  6. Nov 12, 2009 #5

    LCKurtz

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    Science Advisor
    Homework Helper
    Gold Member

    No, it isn't quite the same. The roots of the second derivative are the values of x where there may but there need not be an inflection point. You must always additionally check that the second derivative actually changes sign at those roots. For example, if

    [tex]f''(x) = (x-2)(x-5)^2[/tex]

    then x = 2 gives location for an inflection point but x = 5 does not.
     
  7. Nov 12, 2009 #6
    oh yeah i didnt notice the - sign mistake :S

    there is a second part to this question which asks to find the points where it concaves up and down.
    the graph of f is only concaving up isnt it?
     
  8. Nov 12, 2009 #7

    Mark44

    Staff: Mentor

    The graph of f is concave up (concaving isn't a word) when f''(x) > 0, and is concave down when f''(x) < 0. For your problem, f''(x) is positive for some x values and negative for other x values.
     
  9. Nov 12, 2009 #8
    this is how the question is constructed:
    "f is concave down on (-infty ,___)U(___, infty) and its concave up on (___,___)"

    since f''(x) = 0 for x = 0 and x = -5/16
    so when you said "f''(x) is positive for some x values and negative for other x values"
    ==> x = 0, -5/16

    so would it be:
    "f is concave down on (-infty ,-5/16)U(0, infty) and its concave up on (-5/16,0)"
    ?
     
  10. Nov 12, 2009 #9

    Mark44

    Staff: Mentor

    Works for me.
     
  11. Nov 12, 2009 #10
    sweet... thanks for your explanation!
     
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