Track / Stage 3 / 26 of 260 #26 of 1964

Problem 26

AMC 10/12, early questions

Algebra Difficulty 3.0 Find the answer

Given the function $f(x) = x^2 + ax + 2$ , where $x \in [-5, 5]$ ,

(1) When $a = -1$ , find the intervals of monotonicity for the function $f(x)$ .

(2) If the function $f(x)$ is increasing on $[-5, 5]$ , find the range of values for $a$ .

Your answer

A number or a short expression. Spacing, $ signs and \frac vs / are all fine.

Official solution

(1) When $a = -1$ , since the function $f(x) = x^2 - x + 2 = \left(x - \frac{1}{2}\right)^2 + \frac{7}{4}$ , and $x \in [-5, 5]$ ,

the interval of decrease is $[-5, \frac{1}{2}]$ , and the interval of increase is $\left(\frac{1}{2}, 5\right]$ .

(2) If the function $f(x)$ is increasing on $[-5, 5]$ , then for the quadratic function $f(x) = x^2 + ax + 2$ , its axis of symmetry $x = -\frac{a}{2} \leq -5$ ,

solving this yields $a \geq 10$ . Therefore, the range of values for $a$ is $\boxed{[10, +\infty)}$ .

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