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Optimization

AP Calculus AB· difficulty 3/5

From a 12×1212 \times 12 sheet, square corners of side xx are cut and the sides folded up to form an open box. The volume V(x)=x(12−2x)2V(x) = x(12 - 2x)^2 is maximized at x=x =

  • A

    44

  • B

    22

    check_circle
  • C

    11

  • D

    33

Explanation

V′(x)=(12−2x)2+x⋅2(12−2x)(−2)=(12−2x)(12−6x)=0V'(x) = (12-2x)^2 + x \cdot 2(12-2x)(-2) = (12-2x)(12-6x) = 0. x=6x = 6 (degenerate) or x=2x = 2. Max at x=2x = 2.

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