Question

How do you determine where the function is increasing or decreasing, and determine where relative maxima and minima occur for `f(t) = 6t + 1/t`?

Answer 1

`f (t)= 6t+1/t`
`f'(t)= 6-1/(t^2)`
`f'(t)>0` for `t>1/(sqrt6) or t <-1/(sqrt6)`
`:. f (t)` increases for `t>1/(sqrt6) or t <-1/(sqrt6)`
And `f (t)` decreases for `-1/ sqrt6<` `t` `<` `1/sqrt6`
Now as the derivative changes from positive to negative at `t=-1/sqrt6` therefore it is a point of Maximum
Also as derivative changes from negative to positive at `t= 1/ sqrt6` therefore it is a point of minimum

Answer 2

The function is increasing when `x in (-oo,-1/sqrt6)uu(1/sqrt6,+oo)`
The function is decreasing when `x in (-1/sqrt6,0)uu(0,1/sqrt6)`
See below

Explanation:

Let's calculate the first derivative

`f(t)=6t+1/t`

The domain of `f(t)` is `D_(f(t))=RR-{0}`

`f'(t)=6-1/t^2`

The critical points are when `f'(t)=0`, that is

`6-1/t^2=0`

`6=1/t^2`

`t^2=1/6`

`t=+-1/sqrt6`

Let's build a chart

`color(white)(aaaa)` `t` `color(white)(aaaa)` `-oo` `color(white)(aaaa)` `-1/sqrt6` `color(white)(aaaaaaaa)` `0` `color(white)(aaaaaa)` `1/sqrt6` `color(white)(aaaa)` `+oo`

`color(white)(aaaa)` `f'(t)` `color(white)(aaaaaa)` `+` `color(white)(aaaaaa)` `-` `color(white)(aaaa)` `||` `color(white)(aaaa)` `-` `color(white)(aaaa)` `+`

`color(white)(aaaa)` `f(t)` `color(white)(aaaaaa)` `↗` `color(white)(aaaaaa)` `↘` `color(white)(aaaa)` `||` `color(white)(aaaa)` `↘` `color(white)(aaaa)` `↗`

Let's calculate the second derivative

`f''(t)=2/t^3`

There is no point of inflection.

`f''(-1/sqrt6)=-29.39`, as `f''(-1/sqrt6)<0`, this a local maximum at `(-0.408,-4.899)`

`f''(1/sqrt6)=29.39`, as `f''(1/sqrt6)>0`, this a local minimum at `(0.408,4.899)`

graph{6x+1/x [-41.1, 41.08, -20.54, 20.57]}