1999 CMO 第 2 题

Determine the maximum value of $\lambda$ such that if $f(x) = x^3 +ax^2 +bx+c$ is a cubic polynomial with all its roots nonnegative, then $$f(x)\geq\lambda(x -a)^3$$ for all $x\geq0$ . Find the equality condition.

确定 $\lambda$ 的最大值，使得如果 $f(x) = x^3 +ax^2 +bx+c$ 是所有根均非负的三次多项式，则 $$f(x)\geq\lambda(x -a)^3$$ 对于所有 $x\geq0$ 。求出相等条件。