# Find the local minimum and local maximum of the function f(x) = 6x + 6x^(-1).

## Question:

Find the local minimum and local maximum of the function {eq}f(x) = 6x + 6x^{-1} {/eq}.

## Maxima Or Minima:

When we get the first derivative of a function and equate the result to zero, solving for the value/s of x will give us the critical point/s of the function. If there are two critical points then it can be maxima or minima or both. If there is only one critical point then it could it either maxima or minima point but not both.

The given curve is :

{eq}f(x) = 6x + 6x^{-1}\\ {/eq}

We find the first order derivative, to find the critical point as follows:

{eq}f'(x)=6-\frac{6}{x^2}=0\\ \Rightarrow x=1,\:x=-1\\ {/eq}

Thus at {eq}x=-1 {/eq}, we have:

{eq}f(-1) = 6(-1) + 6(-1)^{-1}\\ =-12\\ \text{Local Minima}\\ {/eq}

Also, at {eq}x=1 {/eq}, we have:

{eq}f(1) = 6(1) + 6(1)^{-1}\\ =12\\ \text{Local Maxima}\\ {/eq}