Find the global maximum and minimum values of the function on the given interval. State as ordered pairs.
$y=x+\frac{2}{x};[0.25,3]$

• A. Global maximum at $\left(0.25,8.25\right)$, Global minimum at $\left(0,2\right)$
• B. Global maximum at $\left(0.25,8.25\right)$, Global minimum at $\left(\sqrt2,2\sqrt2\right)$
• C. Global maximum at $\left(0.25,8.25\right)$, Global minimum at $\left(3,3.67\right)$
• D. Global maximum at $\left(\sqrt2,2\sqrt2\right)$, Global minimum at $\left(-\sqrt2,-2\sqrt2\right)$

Answer: (B)

Verified step by step guidance

1. Identify the function to analyze: \( y = x + \frac{2}{x} \) and the interval \([0.25, 3]\).
2. Find the derivative of the function \( y = x + \frac{2}{x} \) to determine critical points. The derivative is \( y' = 1 - \frac{2}{x^2} \).
3. Set the derivative equal to zero to find critical points: \( 1 - \frac{2}{x^2} = 0 \). Solve for \( x \) to find the critical points within the interval.
4. Evaluate the function \( y = x + \frac{2}{x} \) at the critical points and at the endpoints of the interval \( x = 0.25 \) and \( x = 3 \).
5. Compare the function values obtained in the previous step to determine the global maximum and minimum values, and state them as ordered pairs.