Find the absolute maximum and absolute minimum values of \( f \) on the given interval. \[ f(x)=x+\frac{4}{x},[0.2,8] \] absolute minimum value absolute maximum value

Below is the step‐by‐step solution. --- ### 1. Given Function and Interval We have \[ f(x)=x+\frac{4}{x} \] on the closed interval \([0.2,\,8]\). --- ### 2. Find the Critical Points To find the critical points, compute the derivative: \[ f'(x)= \frac{d}{dx}\left(x+\frac{4}{x}\right) = 1 - \frac{4}{x^2}. \] Set \( f'(x) \) equal to zero: \[ 1 - \frac{4}{x^2} = 0. \] Solve for \( x \): \[ 1 = \frac{4}{x^2} \quad \Longrightarrow \quad x^2 = 4 \quad \Longrightarrow \quad x = 2 \text{ (since } x>0 \text{)}. \] Thus, \( x=2 \) is a critical point in the interval. --- ### 3. Evaluate the Function at Critical Points and Endpoints We now compute \( f(x) \) at the critical point \( x=2 \) and at the endpoints \( x=0.2 \) and \( x=8 \). - **At \( x=2 \):** \[ f(2)=2+\frac{4}{2}=2+2=4. \] - **At \( x=0.2 \):** \[ f(0.2)=0.2+\frac{4}{0.2}=0.2+20=20.2. \] - **At \( x=8 \):** \[ f(8)=8+\frac{4}{8}=8+0.5=8.5. \] --- ### 4. Determine the Absolute Maximum and Minimum From the evaluations: - \( f(0.2) = 20.2 \) - \( f(2) = 4 \) - \( f(8) = 8.5 \) Thus, the **absolute minimum value** is \(\boxed{4}\) at \( x = 2 \), and the **absolute maximum value** is \(\boxed{20.2}\) at \( x = 0.2 \). --- ### Final Answer - **Absolute minimum value:** \(4\) (occuring at \( x=2 \)) - **Absolute maximum value:** \(20.2\) (occuring at \( x=0.2 \))