Answer:
C. (2, 2, 4)
Step-by-step explanation:
You want a set of coefficients (a, b, c) such that f(x)=x³+ax²+bx+c is a monotonic function.
Monotonic
The function will be monotonic if its slope does not change sign. The slope is given by the derivative ...
f'(x) = 3x² +2ax +b
Discriminant
This quadratic will not change sign if the number of roots is 0 or 1. That is, its discriminant must be non-positive.
d = (2a)² -4(3)(b) = 4(a² -3b)
For d ≤ 0, we require ...
a² -3b ≤ 0
b ≥ a²/3
Choices
For the offered values of 'a', we require ...
a = 1, b ≥ 1/3 . . . . . . eliminates choice B
a = 2, b ≥ 4/3 . . . . . eliminates choices A and D
Possible values of (a, b, c) are (2, 2, 4), matching choice C.
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Additional comment
The functions are graphed in the attachment. The black curve is monotonic, and corresponds to choice C.
