匿名
匿名 發問於 科學及數學數學 · 7 年前

Calculus---derivative

Explain why the cubic polynomial function

f(x)=ax^(3) + bx^(2) + cx +d (a unequal 0)

can have either two, one, or no critical points on the real line.

Produce examples that illustrate each of the three cases.

3 個解答

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  • Wen
    Lv 5
    7 年前
    最愛解答

    觀察critical的特性,會與f'(x)的 root有關

    f'是一個二次函數,所以它的root會有相異根、重根、無實根。

    2013-10-16 12:31:35 補充:

    f'(x) = 3ax^2 + 2bx + c

    if f'(t) = 0 or not exist, then we call that t is critical point.

    Since f'(x) is polynomial, so f'(x) is exist on R

    => no one t such that f'(t) is not exist.

    consider f'(x) = 0

    i.e ax^2 + 2bx + c = 0

    x = (-2b ± √D)/2a, where D = (4b^2 - 4ac)

    if f have two critical pts, the mean is that roots of f'(x) are distinct.

    =>D > 0

    if f has one critical pt, the mean is that roots of f'(x) are same

    =>D = 0

    if f has no critical pt, the mean is that f'(x) hsa no real root

    =>D < 0

    Examples:two: Let b=2, a = c = 1 => D = 12 > 0,d is arb number, we can let d = 0

    one: Let a = b = a = 1 => D = 0, d is arb number, we can let d = 3

    no: let b = 0, a = c = 1 => D = -4 <0, d is arb number, we can let d = -9

    Done.

  • 匿名
    7 年前

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  • 7 年前

    所以講到尾要同佢講 fundamental theorem of algebra.

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