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Second Derivative Test

AP Calculus AB· difficulty 4/5

f(x)=x4−4x3+16f(x) = x^4 - 4x^3 + 16 has a local minimum at

  • A

    x=2x = 2

  • B

    x=−1x = -1

  • C

    x=0x = 0

  • D

    x=3x = 3

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Explanation

f′=4x3−12x2=4x2(x−3)f' = 4x^3-12x^2 = 4x^2(x-3). Critical: x=0,3x=0,3. f′′=12x2−24xf''=12x^2-24x. f′′(3)=108−72=36>0f''(3)=108-72=36>0 → local min. f′′(0)=0f''(0)=0 inconclusive but sign of f′f' doesn't change at 0.

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