Question

find the absolute maximum and minimum values of the function over the indicated interval, and indicate the x-values at which they occur
$f(x) = x + \frac{9}{x}, 3,20$

the absolute maximum value is \\(\square\\) at \\(x = \square\\)
(use a comma to separate answers as needed )

the absolute minimum value is \\(\square\\) at \\(x = \square\\)
(use a comma to separate answers as needed )

Explanation:

Step1: Find critical points

Compute derivative: $f'(x) = 1 - \frac{9}{x^2}$. Set to 0: $1 - \frac{9}{x^2} = 0 \Rightarrow x^2 = 9 \Rightarrow x = 3$ (since $x=-3$ not in [3,20]).

Step2: Evaluate f at critical point and endpoints

At $x=3$: $f(3)=3+\frac{9}{3}=6$; at $x=20$: $f(20)=20+\frac{9}{20}=20.45$.

Step3: Identify max and min

Compare values: 6 (min), 20.45 (max).

Answer:

The absolute maximum value is 20.45 at x=20
The absolute minimum value is 6 at x=3