Find the absolute maximum and minimum values of the function over the indicated interval, and indicate the $x$-values at which they occur.
$$f(x)=x+\frac{1}{x};[1,25]$$
The absolute maximum value is $\frac{626}{25}$ at $x=25$.
(Use a comma to separate answers as needed.)
The absolute minimum value is at $x=$ (Use a comma to separate answers as needed.)

Step 1 ：The function given is $f(x)=x+\frac{1}{x}$ and we are to find the absolute maximum and minimum values of the function over the interval [1,25].

Step 2 ：The function is continuous and differentiable on the interval [1,25]. Therefore, the absolute maximum and minimum values of the function occur either at the endpoints of the interval or at critical points in the interval.

Step 3 ：To find the critical points, we need to find the derivative of the function and set it equal to zero. The derivative of the function $f(x)=x+\frac{1}{x}$ is $f^{\prime }(x)=1-\frac{1}{x^2}$.

Step 4 ：Setting this equal to zero gives $x^2=1$, so $x=\pm 1$. However, only $x=1$ is in the interval [1,25].

Step 5 ：We then evaluate the function at the endpoints of the interval and at the critical point to find the absolute maximum and minimum values.

Step 6 ：At $x=1$, $f(x)=1+\frac{1}{1}=2$.

Step 7 ：At $x=25$, $f(x)=25+\frac{1}{25}=\frac{626}{25}$.

Step 8 ：Comparing these values, we find that the absolute maximum value is $\frac{626}{25}$ at $x=25$ and the absolute minimum value is $2$ at $x=1$.

Step 9 ：$\boxed{{\displaystyle \text{Final Answer: The absolute maximum value is }\frac{626}{25}\text{ at }x=25\text{. The absolute minimum value is }2\text{ at }x=1.}}$