Question

How do you determine where the function is increasing or decreasing, and determine where relative maxima and minima occur for `f (x) = x + 4/x`?

Answer 1

You determine studying the derivative.

Explanation:

You have to evaluate the derivative of the function.
The derivative is the value of the slope, so when the derivative is positive the function is increasing, when it is negative the function is decreasing and when it is zero you have a maximum, a minimum or an horizontal flex.
So let's calculate the derivative

`(df)/dx=1-4/x^2`

Its graph is in figure
graph{1-4/x^2 [-10.17, 9.83, -4.92, 5.08]}

When `x<-2` or `x>2` the derivative is positive, so the function is increasing. When `-2<x<2` the derivative is negative, so the function is decreasing, when `x=-2` the function passes from the increasing to decreasing, so it is a maximum, while when `x=2` the function passes from decreasing to increasing, then the function has a minimum.

To verify this we can see the plot of `f(x)`.

graph{x+4/x [-20.17, 19.83, -9.92, 10.08]}