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Math Help - How to find the maximum point of a graph

  1. #1
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    Exclamation How to find the maximum point of a graph

    Question: Find the maximum y value of the graph between 0 ≤ x ≤ 3

    Equation:
    f(x) = x(8-2x)(6-2x)

    Answer:
    24.26


    How would you find the maximum y value of this graph? Is it not like a parabola so when I tried finding the midpoint and then using the quadratic formula it didn't give me the highest value. Please explain the process and show your work. Thanks.
    Last edited by florx; May 2nd 2010 at 02:54 PM.
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  2. #2
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    Equation:
    f(x) = x(8-2x)(6x-2x)??
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  3. #3
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    Oops typo.

    Corrected:
    f(x) = x(8-2x)(6-2x)
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  4. #4
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    Equation:
    f(x) = x(8-2x)(6-2x) = 4x(4-x)(3-x) = 4x^{3}-28x^{2}+48x

    at the turning points

    d/dx = 0
    d/dx = 12x^{2}-56x+48
    roots are (7-sqr(13))/3 and (7+sqr(13))/3
    until now all we know is these are turning points,
    The value f (x) will tell us whether the point is a maximum or a minimum or a point of inflection

    differentiate again

    f''(x)=24x-56= 8(3x-7)

    f''((7+sqr(13))/3) = positive, hence it is min
    f''((7-sqr(13))/3) = negative, so this is max
    f((7-sqr(13))/3) = 24.26

    Answer:
    24.26
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  5. #5
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    you also need to check the global extrema, where the graph gets cut off e.g. at where x=3 and x=0 however in this case they are less than this value
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