Question

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.

Answer 1

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