If the solve the problem
Question:

$f(x)=\frac{x}{2}+\frac{2}{x}, x>0$

Solution:

Given: $f(x)=\frac{x}{2}+\frac{2}{x}$

$\Rightarrow f^{\prime}(x)=\frac{1}{2}-\frac{2}{x^{2}}$

For the local maxima or minima, we must have

$f^{\prime}(x)=0$

$\Rightarrow \frac{1}{2}-\frac{2}{x^{2}}=0$

$\Rightarrow \frac{1}{2}=\frac{2}{x^{2}}$

$\Rightarrow x^{2}=\pm 2$

Since $x>0, f^{\prime}(x)$ changes from negative to positive when $x$ increases through $2 .$ So, $x=2$ is a point of local minima.

The local minimum value of $f(x)$ at $x=2$ is given by

$\frac{2}{2}+\frac{2}{2}=2$