The maximum value of $y=x^2-6x+9$ obtained when $x$ varies over the interval $2$ to $5$ is $9$.
To find the maximum value of a function, we can find the critical points and then compare the function’s value at the critical points and the endpoints of the interval.
The derivative of $y=x^2-6x+9$ is $y’=2x-6$. The critical points of $y$ are the points where $y’=0$. Solving $y’=0$, we find that the critical point is $x=3$.
Since $y’$ is a polynomial, it is defined for all real numbers. Therefore, the critical point $x=3$ is within the interval $2$ to $5$.
To compare the function’s value at the critical point and the endpoints of the interval, we can evaluate $y$ at $x=2$, $x=3$, $x=4$, and $x=5$.
We find that $y(2)=1$, $y(3)=9$, $y(4)=7$, and $y(5)=1$.
Therefore, the maximum value of $y$ obtained when $x$ varies over the interval $2$ to $5$ is $9$.