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21 July, 07:14

For what value of x will the function f (x) = - 3 (x - 10) (x - 4) have a maximum value? Find the maximum value.

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  1. 21 July, 08:20
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    Hello!

    To find the maximum value of the function f (x) = - 3 (x - 10) (x - 4), the easiest way is to find the vertex using the formula: x = - b/2a.

    Firstly, we need to simplify f (x).

    f (x) = - 3 (x - 10) (x - 4)

    f (x) = - 3 (x² - 14x + 40)

    f (x) = - 3x² + 42x + - 120

    Since the equation f (x) is now simplified to standard form, we can find the vertex.

    a = - 3, b = 42, and c = - 120

    x = - (42) / 2 (-3) = - 42/-6 = 7

    Then, we substitute 7 into the the function f (x) = - 3 (x - 10) (x - 4), or

    f (x) = - 3x² + 42x + - 120, to find the y-value of the vertex.

    f (x) = - 3 (7 - 10) (7 - 4)

    f (x) = - 3 (-3) (4)

    f (x) = 27

    The vertex of f (x) is (7, 27).

    Therefore, the maximum x-value for the function f (x) is 7.
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