For real $x$, the greatest value of $1/(x^2-4x+7)$ is:

Complete the square in the denominator: $x^2-4x+7=(x-2{)}^2+3$, which is always at least 3 and equals 3 at $x=2$. A fraction with a fixed numerator is largest when its denominator is smallest, so the greatest value is 1/3. (3 is the least value of the denominator, not of the fraction; 1/7 is the value at $x=0$; 1/4 comes from writing the denominator as $(x-2{)}^2+4$.) At $x=2$ the expression is $1/(4-8+7)=1/3$, and at $x=3$ it is 1/4.