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Let g be the function give by g(x) = x^4 - 4x^3 + 6x^2 – 4x + k where k is constant.
A.On what intervals is g increasing? Justify your answer.
B.On what interval is g concave upward? Justify your answer.
C.Find the value of k for which g has 5 as its relative minimum. Justify your answer.

  • calculus -

    g = x^4 - 4x^3 + 6x^2 – 4x + k
    g' = 4x^3 - 12x^2 + 12x - 4 = 4(x-1)^3
    g'' = 12x^2 - 24x + 12 = 12(x-1)^2

    g increasing where g' > 0: x>1
    g concave up where g'' > 0: all real x

    g(x) = (x-1)^4 + (k-1)
    so, if g(1) = 5, k=6

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