Question

The function `f: f(x)=x^3+6x+2` is increasing when `x` belong to .......?

Answer 1

The given function is monotonically increasing when x belongs to the set of real numbers(`R`)

Explanation:

`f(-2) = -12`
`f(-1) = -5`
`f(0) = 2`
`f(1) = 9`

(Also, `fprime(x) = 3x^2 + 6`, which is always positive. So `f(x)`is strictly increasing.)

Clearly, f(x) increases as x increases.
graph{y = x^3 + 6x + 6 [-58.5, 58.5, -29.25, 29.3]}

Answer 2

The function is increasing `x in RR`

Explanation:

The function is

`f(x)=x^3+6x+2`

The derivative is

`f'(x)=3x^2+6`

The critical points are when `f'(x)=0`

As

`3x^2+6>0`

`AA x in RR, f'(x)>0`

Therefore,

The function is increasing `x in RR`

The second derivative is

`f''(x)=6x`

`f''(x)=0` when `x=0`

There is a point of inflection at `(0,2)`

graph{x^3+6x+2 [-2.48, 2.995, 0.721, 3.461]}