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

May 1, 2018

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

#### Explanation:

$f \left(- 2\right) = - 12$
$f \left(- 1\right) = - 5$
$f \left(0\right) = 2$
$f \left(1\right) = 9$

(Also, $f p r i m e \left(x\right) = 3 {x}^{2} + 6$, which is always positive. So $f \left(x\right)$is strictly increasing.)

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

May 1, 2018

The function is increasing $x \in \mathbb{R}$

#### Explanation:

The function is

$f \left(x\right) = {x}^{3} + 6 x + 2$

The derivative is

$f ' \left(x\right) = 3 {x}^{2} + 6$

The critical points are when $f ' \left(x\right) = 0$

As

$3 {x}^{2} + 6 > 0$

$\forall x \in \mathbb{R} , f ' \left(x\right) > 0$

Therefore,

The function is increasing $x \in \mathbb{R}$

The second derivative is

$f ' ' \left(x\right) = 6 x$

$f ' ' \left(x\right) = 0$ when $x = 0$

There is a point of inflection at $\left(0 , 2\right)$

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