# How do you find intercepts, extrema, points of inflections, asymptotes and graph y=3x^4+4x^3?

Mar 14, 2017

See explanation below

#### Explanation:

$y = 3 {x}^{4} + 4 {x}^{3}$

This is a polynomial function

The domain of $y$ is ${D}_{y} = \mathbb{R}$

We calculate the first and second derivatives

$\frac{\mathrm{dy}}{\mathrm{dx}} = 12 {x}^{3} + 12 {x}^{2}$

The critical points are when

$\frac{\mathrm{dy}}{\mathrm{dx}} = 0$

$12 {x}^{3} + 12 {x}^{2} = 0$

$12 {x}^{2} \left(x + 1\right) = 0$

Therefore,

$x = 0$ and $x = - 1$

When $x = 0$, $\implies$, $y ' ' = 0$, $\implies$, point of inflexion

When $x = - 1$, $\implies$, $y ' ' > 0$, a minimum

We build a sign chart

$\textcolor{w h i t e}{a a a a}$$x$$\textcolor{w h i t e}{a a a a}$$- \infty$$\textcolor{w h i t e}{a a a a}$$- 1$$\textcolor{w h i t e}{a a a a}$$0$$\textcolor{w h i t e}{a a a a a}$$+ \infty$

$\textcolor{w h i t e}{a a a a}$$x + 1$$\textcolor{w h i t e}{a a a a a}$$-$$\textcolor{w h i t e}{a a a a}$$+$$\textcolor{w h i t e}{a a a a}$$+$

$\textcolor{w h i t e}{a a a a}$$y$$\textcolor{w h i t e}{a a a a a a a a a}$↘$\textcolor{w h i t e}{a a a a}$↗$\textcolor{w h i t e}{a a a a}$↗

We calculate the second derivative

$\frac{{d}^{2} y}{\mathrm{dx}} ^ 2 = 36 {x}^{2} + 24 x$

$\frac{{d}^{2} y}{\mathrm{dx}} ^ 2 = 0$

$36 {x}^{2} + 24 x = 0$

$12 x \left(3 x + 2\right) = 0$

Therefore,

$x = 0$ and $x = - \frac{2}{3}$

These are the points of inflexions

We construct a chart

$\textcolor{w h i t e}{a a a a}$$I n t e r v a l$$\textcolor{w h i t e}{a a a a}$]-oo,-2/3[$\textcolor{w h i t e}{a a a a}$]-2/3,0[$\textcolor{w h i t e}{a a a a}$]0,+oo[

$\textcolor{w h i t e}{a a a a}$$s i g n \left(y ' '\right)$$\textcolor{w h i t e}{a a a a a a a a a}$$+$$\textcolor{w h i t e}{a a a a a a a a a a}$$-$$\textcolor{w h i t e}{a a a a a a a a}$$+$

$\textcolor{w h i t e}{a a a a}$$y$$\textcolor{w h i t e}{a a a a a a a a a a a a a a a a a}$$\cup$$\textcolor{w h i t e}{a a a a a a a a a a}$$\cap$$\textcolor{w h i t e}{a a a a a a a a}$$\cup$

The $\textcolor{red}{I n t e r c e p t s}$ are

$\left(0 , 0\right)$ and

$3 {x}^{4} + 4 {x}^{3} = 0$

${x}^{3} \left(3 x + 4\right) = 0$

$x = 0$ and $x = - \frac{4}{3}$

$\textcolor{red}{\text{limits}}$

${\lim}_{x \to \pm \infty} y = {\lim}_{x \to \pm \infty} 3 {x}^{4} = + \infty$

graph{3x^4+4x^3 [-3.082, 3.075, -1.538, 1.542]}