Question

How do you determine where the function is increasing or decreasing, and determine where relative maxima and minima occur for `f(x)= 2x+3/x`?

Answer 1

Evaluate the first derivative of the function:

`f'(x) = d/dx (2x+3/x) = 2-3/x^2 = (2x^2-3)/x^2`

as the denominator is always positive, the sign of `f'(x)` is the sign of the numerator, which means that `f'(x) < 0` when:

`2x^2-3 <0`

`absx < sqrt(3/2)`

So the function `f(x)` is increasing in `(-oo, -sqrt(3/2))`, decreasing in `(-sqrt(3/2), sqrt(3/2))` and again increasing in `(sqrt(3/2),+oo)`. For `x=-sqrt(3/2)` the function has a local maximum and for `x=sqrt(3/2)` it has a local minimum.

graph{2x+3/x [-20, 20, -10, 10]}