Question

Find the local maxima and local minima, if any of the given functions. Find also the local maximum and local minimum values, as the case may be:

$g\left(x\right)=\frac{x}{2}+\frac{2}{x},x>0$

Answer 1

Maximum or minimum can be seen by using derivatives.

Step 1: First find first derivative of the function

Step 2: Put it equal to zero and find $x$ were first derivative is zero

Step 3: Now find second derivative

Step 4: Put $x$ for which first derivative was zero in equation of second derivative

Step 5: If second derivative is greater than zero then function takes minimum value at that $x$ and if second derivative is negative then function will take maximum value at that $x$. If Second derivative is zero them it means that this is the point of inflection.

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

Putting this equal to zero, we get

$g^{{}^{\prime }}\left(x\right)=\frac{1}{2}-\frac{1}{x^2}=0$
$\Rightarrow x=\pm 2$.

Please note that $-2$ is not in the domain of given function so it is of no use to us
Now let's see the double derivative of this function.
$f^{{}^{\prime \prime }}\left(x\right)=\frac{4}{x^3}$
At $x=2$
$f^{{}^{\prime \prime }}\left(2\right)=\frac{4}{2^3}=\frac{1}{2}$
Which is positive, so function will take minimum value at $x=2$
Minimum value is given by
$f\left(x\right)=\frac{2}{2}+\frac{2}{2}=2$