Question

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

Answer 1

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

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

$$\text{ For the local maxima or minima, we must have }$$

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

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

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

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

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