Chinese Math Competition Problem

Calculus Level 3

$f(x)=x^{3}-(3-2a)x^{2}+ax$ , ( $x \in \mathbb{R}$ ), is a monotonic function, where $a$ is a parameter. The value range of $a$ is then?

[4,6] [3/4,3] [2/3,3/4] (3/4,3) (4,6)

1 solution

Loillipop Cuber
Feb 13, 2018

f(x) is monotonic, and the coefficient of x^3 is positive, so f is increasing. The derivative of f(x) is 3x^2-(6-4a)x+a, and since f is always increasing or zero at one instance, the discriminant of f’(x) must less than or equal to zero. Setting f’(x) less than or equal to zero we get the interval [3/4,3].

Chinese Math Competition Problem

1 solution

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