Problem - 3652
If real number $x$ satisfies $x^4 - 2x^3 -7x^2 + 8x +12\le 0$, find the max value of $|x+\frac{4}{x}|$
The given condition can be first factorized and then solved as $$(x-3)(x-2)(x+1)(x+2)\le 0 \implies x\in (-2, -1)\cup(2,3)$$
The target function $|x+\frac{4}{x}|$ monotonically increase when $x < 0$ and monotonically decrease when $x > 0$. Hence we only need to compare its value at $x=-1$ and $x=2$ and find its maximum value is $5$.