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Homework Help: Fuction and Graph

  1. Jan 22, 2007 #1
    1. The problem statement, all variables and given/known data
    The function f is defined as
    f(x)=3x^5-10x^3+1
    1) Determine the maximum and minimum points, as well the point of inflection of the graph f.

    2. Relevant equations
    All the differentiation equation.


    3. The attempt at a solution

    I found the maximum point and minimum point, but I had some trouble with the inflection point.

    So....
    f''(x)=60x^3-60x^2
    f''(x)=0,
    60x^3-60x^2=0
    x=0, x=1

    f'''(0)=0

    Hence the only inflection point is (1,-6)

    But the answer given is (0,1),(1,-6)and (-1,8). Is the answer correct?
     
  2. jcsd
  3. Jan 22, 2007 #2
    Not sure this should be in this topic as it is calculus not precalculus.

    Have another look at your differentiation, in particular check the term -60x^2 in your expression for f''(x).
     
  4. Jan 22, 2007 #3
    This second derivative is wrong. Calculate it again.

    Your approach is correct though.

    marlon
     
  5. Jan 22, 2007 #4

    HallsofIvy

    User Avatar
    Science Advisor

    f"(x)= 60x3- 60x= 60x(x2- 1)= 0
    for x= 0, 1, and -1.

    One definition of "inflection point" is that the second derivative changes sign there. Yes, it is true that since f '''(0)= 0, the second derivative might NOT change signs there. For example, if f ''= x2, then while f''(0)= 0, f'' does not change signs there. But f'''(0)= 0 does not mean it CAN'T change signs there. For example, if f ''= x3, then, again, f'''(x)= 3x2 so f'''(0)= 0 but x3 does change signs there.

    In this particular case, f''(1/2)= 60/8- 120< 0 while f''(-1/2)= -60/8+120> 0 so f'' does change signs at 0.
     
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