Question

For values of $x$ less than $1$ but greater than $-4,$ the expression

$\frac{x^2-2x+2}{2x-2}$

has:

no maximum or minimum value

a minimum value of $1$

a maximum value of $1$

a minimum value of $-1$

a maximum value of $-1$

Answer: E

Concepts:AM-GM Inequality inequality optimization

Difficulty rating: 1830

Small Hint:

Set $t=x-1,$ which is negative on the given interval

Big Hint:

Rewrite the expression as $\tfrac12(t+\frac{1}{t})$ and use the negative- $t$ form of AM-GM

Solution:

Set $t=x-1.$ Then $-5\lt t\lt0,$ and the expression becomes $\frac{t^2+1}{2t} =\frac12\left(t+\frac1t\right).$ For $t\lt0,$ $t+\frac{1}{t}\le-2,$ with equality when $t=-1.$ That value is allowed and corresponds to $x=0.$ Hence the expression is at most $-1,$ and its maximum is $-1.$

Thus, the correct answer is E.

1958 AMC 12 Problem 46

Answer: E

Difficulty rating: 1830

Small Hint:

Big Hint:

Solution:

Problem 46 in Other Years