Question

The number of ordered pairs \((a,b)\), such that the function \(f(x)=4x^{3}-(3a+2b)x^{2}+2abx\) is decreasing in \(\left[\frac32,\frac53\right]\) and increasing in \(\mathbb{R}-\left[\frac32,\frac53\right]\), is ________.

Hint:

The derivative must be \(\le0\) exactly on \(\left[\frac32,\frac53\right]\), so its roots are \(\frac32\) and \(\frac53\); compare coefficients to get \(3a+2b\) and \(ab\).

Answer 1

Correct Answer: 2

\(f'(x)=12x^2-2(3a+2b)x+2ab\). For \(f\) to decrease exactly on \(\left[\frac32,\frac53\right]\) and increase elsewhere, \(f'(x)\) must be an upward parabola with roots \(\frac32\) and \(\frac53\): \(f'(x)=12\left(x-\frac32\right)\left(x-\frac53\right)\).
Sum of roots: \(\frac{2(3a+2b)}{12}=\frac32+\frac53=\frac{19}{6}\), so \(3a+2b=19\).
Product of roots: \(\frac{2ab}{12}=\frac32\cdot\frac53=\frac52\), so \(ab=15\).
Substituting \(b=\frac{19-3a}{2}\): \(a(19-3a)=30\), i.e. \(3a^2-19a+30=0\), \((3a-10)(a-3)=0\).
So \(a=3,\ b=5\) or \(a=\frac{10}{3},\ b=\frac92\). Both satisfy \(ab=15\).
Number of ordered pairs = 2.
Answer: 2