Track / Stage 4 / 283 of 340 #543 of 2000

Problem 543

AMC 12 late, AIME early

Algebra Difficulty 5.0 Find the answer

1. For $-4<x<1$ , regarding the expression $\frac{x^{2}-2 x+2}{2 x-2}$
A. No maximum or minimum value
B. Has a minimum value of 1
C. Has a maximum value of 1
D. Has a minimum value of -1
E. Has a maximum value of -1

Your answer

Official solution

1. E Let $f(x)=\frac{x^{2}-2 x+2}{2 x-2}$ , then $f^{\prime}(x)=\frac{x(x-2)}{2(x-1)^{2}}, f^{\prime}(x)=0, x=0,2 . f^{\prime \prime}(x)=\frac{1}{(x-1)^{3}}$ . $f^{\prime \prime}(0)=-1, f^{\prime \prime}(2)=1, f(0)=-1, f(2)=1$ , but $-4<x<1$ . Therefore, the answer is E.

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