find the critical numbers of f(x) = X^4 (x – 2)³?

applications of differentiation

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  • 1 decade ago
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    Given f(x) = x^4(x-2)³

    Take the derivative of f(x) using product rule:

    f '(x) = 4x³ * (x - 2)³ + 3(x-2)² * x^4

    Simplify (I will let you do this since this only algebra)

    f '(x) = 7x^6 - 36x^5 + 60x^4 - 32x^3

    set f ' (x) = 0

    7x^6 - 36x^5 + 60x^4 - 32x^3 = 0

    Factor out x³

    x³(7x³ - 36x² + 60x - 32) = 0

    x³ = 0 or 7x³ - 36x² + 60x - 32 = 0

    x = 0 <==== That's one of your critical numbers

    Now, we have to solve 7x³ - 36x² + 60x - 32 = 0

    Solving the above equation gives (I will let you do that, you can also use mathematical software to find those zeros)

    x = 2 and x = 8/7

    Therefore, the critical numbers are x = 0, 8/7, 2.

    Source(s): LZY8 Math Professor
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