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calculus

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Let q(x)=(x^4)(e^-x). Calculate q'(x). Use q'(x) to determine whether q(5)<q(6) or q(5)>q(6) (do not use a calculator). Explain your reasoning.

  • calculus -

    use the product rule to calculate
    q'(x)=4x3e^(-x)+x4*(-e^(-x)
    = (4-x)x^3*e^(-x)
    Since e^(-x)>0 on ℝ, this means
    that q'(x) has zeroes at x=0 and x=4 only.

    q'(5)=-125e^(-5)<0
    which demonstrates that q(x) is strictly monotonically decreasing on [4,∞) and therefore...

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