Find the local maxima and minima of $g(x) = \frac{x}{2} + \frac{2}{x}, \, x 0 $.

Q: Find the local maxima and minima of $g(x) = \frac{x}{2} + \frac{2}{x}, \, x > 0 $.

Explanation: \[ g'(x) = \frac{1}{2} - \frac{2}{x^2} \Rightarrow g'(x) = 0 \Rightarrow \frac{1}{2} = \frac{2}{x^2} \Rightarrow x^2 = 4 \Rightarrow x = 2 \, (\text{since } x > 0) \] \[ g''(x) = \frac{4}{x^3} \Rightarrow g''(2) = \frac{4}{8} = \frac{1}{2} > 0 \Rightarrow \text{Local minimum} \Rightarrow g(2) = 1 + 1 = 2 \]

Question

Find the local maxima and minima of $g(x) = \frac{x}{2} + \frac{2}{x}, \, x > 0 $.

Accepted Answer

Explanation: $$
g'(x) = \frac{1}{2} - \frac{2}{x^2}
\Rightarrow g'(x) = 0 \Rightarrow \frac{1}{2} = \frac{2}{x^2}
\Rightarrow x^2 = 4 \Rightarrow x = 2 \, (\text{since } x > 0)
$$
$$
g''(x) = \frac{4}{x^3}
\Rightarrow g''(2) = \frac{4}{8} = \frac{1}{2} > 0 \Rightarrow \text{Local minimum}
\Rightarrow g(2) = 1 + 1 = 2
$$

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