Find the local maxima and local minima, if any, of the following functions. Find also the local maximum and the local minimum values as the case may be: $g (x) = \frac{x}{2} + \frac{2}{x}, x > 0$

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Detailed Solution

g (x) = \frac{x}{2} + \frac{2}{x}

g ` (x) = \frac{1}{2} - \frac{2}{x^{2}}

Put

g ` (x) = 0

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

\Rightarrow x = \pm 2

Since x is given as greater than 0, so we get the value of x as 2 only.

g ` ` (x) = \frac{4}{x^{3}}

g ` ` (2) = \frac{4}{2^{3}} = \frac{1}{2} > 0

Therefore 2 is the point of local minima by second derivative test.
Putting x = 2 in g(x) we get,

g (2) = \frac{2}{2} + \frac{2}{2} = 2

Hence, the local minimum value is 2.

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