# Suppose g is a function whose derivative is g'(x)=3x^2+1 Is g increasing, decreasing, or neither at x=0?

May 7, 2018

Increasing

#### Explanation:

$g ' \left(x\right) = 3 {x}^{2} + 1 > 0$ , $\forall$$x$$\in$$\mathbb{R}$ so $g$ is increasing in $\mathbb{R}$ and so is at ${x}_{0} = 0$

Another approach,

$g ' \left(x\right) = 3 {x}^{2} + 1$ $\iff$

$\left(g \left(x\right)\right) ' = \left({x}^{3} + x\right) '$ $\iff$

$g$, ${x}^{3} + x$ are continuous in $\mathbb{R}$ and they have equal derivatives, therefore there is $c$$\in$$\mathbb{R}$ with

$g \left(x\right) = {x}^{3} + x + c$,
$c$$\in$$\mathbb{R}$

Supposed ${x}_{1}$,${x}_{2}$$\in$$\mathbb{R}$ with ${x}_{1} <$${x}_{2}$ $\left(1\right)$

${x}_{1} <$${x}_{2}$ $\implies$ ${x}_{1}^{3} <$${x}_{2}^{3}$ $\implies$ ${x}_{1}^{3} + c <$${x}_{2}^{3} + c$ $\left(2\right)$

From $\left(1\right) + \left(2\right)$

${x}_{1}^{3} + {x}_{1} + c <$${x}_{2}^{3} + {x}_{2} + c$ $\iff$

$g \left({x}_{1}\right) <$$g \left({x}_{2}\right)$ $\to$ $g$ increasing in $\mathbb{R}$ and so at ${x}_{0} = 0$$\in$$\mathbb{R}$