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Determining Concavity

AP Calculus AB· difficulty 4/5

The function f(x)=x4−4x3f(x) = x^4 - 4x^3 has inflection points at

  • A

    x=−1x = -1 and x=3x = 3

  • B

    x=0x = 0 and x=2x = 2

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  • C

    x=2x = 2 only

  • D

    x=0x = 0 only

Explanation

f′=4x3−12x2f'=4x^3-12x^2, f′′=12x2−24x=12x(x−2)f''=12x^2-24x=12x(x-2). Sign of f′′f'' changes at x=0x=0 (positive→negative) and x=2x=2 (negative→positive). Both are inflection points.

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