The greatest value of (1)/(x^2-4x+7) for real x is

The denominator is $x^2-4x+7=(x-2)^2+3\ge3$, with minimum $3$ at $x=2$. The reciprocal of a positive quantity is largest when the quantity is smallest, so the greatest value is $\dfrac13$. The value $\dfrac17$ is the value at $x=0$, and $3$ is the minimum of the denominator, not of the fraction. The fraction is always positive, so no larger value occurs for any real $x$.