If the function $f(x)=a\ln x+\frac{b}{x}+\frac{c}{x^2}$ $(a\neq 0)$ has both a local maximum and a local minimum, then: A. $bc>0$ B. $ab>0$ C. $b^2+8ac>0$ D. $ac<0$
BCD From the problem, the domain of $f(x)$ is $(0,+\infty)$, and $f'(x)=\frac{a}{x}-\frac{b}{x^2}-\frac{2c}{x^3}=\frac{ax^2-bx-2c}{x^3}$. Since $f(x)$ has both a local maximum and a local minimum, $f'(x)$ has two distinct real roots on $(0,+\infty)$. Let $h(x)=ax^2-bx-2c$. Then $h(x)$ has two distinct real roots on $(0,+\infty)$, so $\Delta>0$, $x_1+x_2>0$, and $x_1\cdot x_2>0$.
If the function $f(x)=a\ln x+\frac{b}{x}+\frac{c}{x^2}$ $(a\neq 0)$ has both a local maximum and a local minimum, then:\\
A. $bc>0$\\
B. $ab>0$\\
C. $b^2+8ac>0$\\
D. $ac<0$