Question

The possible values of the expression $\frac{1}{x^2-2x+5}$

• A. (4,$\infty$)
• B. [4,$\infty$)
• C. $(0,\frac{1}{4}]$
• D. $(0,\frac{1}{4})$

Answer 1

The correct option is C

$(0,\frac{1}{4}]$

$x^2-2x+5=x^2-2\cdot 1\cdot x+1-1+5={\left(x-1\right)}^2+4$

As we know that $(x-1{)}^2+4$

$\infty >{\left(x-1\right)}^2+4\ge 4$

Taking reciprocal
$0<\frac{1}{{\left(x-1\right)}^2+4}\le \frac{1}{4}$
$\Rightarrow \frac{1}{x^2-2x+5}$ $\in (0,\frac{1}{4}]$