find the absolute maximum and absolute minimum values of f on the given interval.\n\n$f(x)=x +…

Explanation:

Step1: Find the derivative

Differentiate $f(x)=x + \frac{25}{x}=x + 25x^{-1}$ using the power - rule. The derivative $f'(x)=1-25x^{-2}=1-\frac{25}{x^{2}}$.

Step2: Set the derivative equal to zero

Solve $f'(x) = 0$. So, $1-\frac{25}{x^{2}}=0$. Add $\frac{25}{x^{2}}$ to both sides: $1=\frac{25}{x^{2}}$. Cross - multiply to get $x^{2}=25$, then $x=\pm5$. But since our interval is $[0.2,20]$, we only consider $x = 5$.

Step3: Evaluate the function at critical points and endpoints

Evaluate $f(x)$ at $x = 0.2$, $x = 5$, and $x = 20$.

• When $x=0.2$, $f(0.2)=0.2+\frac{25}{0.2}=0.2 + 125=125.2$.
• When $x = 5$, $f(5)=5+\frac{25}{5}=5 + 5=10$.
• When $x = 20$, $f(20)=20+\frac{25}{20}=20 + 1.25=21.25$.

Answer:

absolute minimum value: $10$ absolute maximum value: $125.2$