Track / Stage 4 / 323 of 340 #583 of 1964

Problem 583

AMC 12 late, AIME early

Algebra Difficulty 5.0 Find the answer

1. Given $f(x)=-\log _{3}\left(x^{2}-a x-a\right)$ is monotonically increasing on $(-\infty, 1-\sqrt{3})$ . Then the range of values for $a$ is $\qquad$ .

Your answer

A number or a short expression. Spacing and $ signs are ignored.

Official solution

$-1.2-2 \sqrt{3} \leqslant a \leqslant 2 \text {. }$

Since $y=\log _{3}\left(x^{2}-a x-a\right)$ is monotonically decreasing on $(-\infty, 1-\sqrt{3})$ , we have:
$\left\{\begin{array}{l} x_{0}=\frac{a}{2} \geqslant 1-\sqrt{3}, \\ (1-\sqrt{3})^{2}-a(1-\sqrt{3})-a \geqslant 0 . \end{array}\right.$

Solving this, we get $2-2 \sqrt{3} \leqslant a \leqslant 2$ .

Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic, difficulty and ordering added by this site.