# How do you sketch the graph by determining all relative max and min, inflection points, finding intervals of increasing, decreasing and any asymptotes given f(x)=x^4-4x^3?

May 5, 2017

See below.

#### Explanation:

$f \left(x\right) = {x}^{4} - 4 {x}^{3}$ $\text{ }$ Domain $\left(- \infty , \infty\right)$

Asymptotes: None. Polynomials don't have (linear) asymptotes.

Intercepts
$x$-intercepts are at solutions to $f \left(x\right) = 0$, so the $x$ intercepts are $0$ and $4$.
$y$-intercepts are at $f \left(0\right)$ so the $y$-intercept is $0$.

Analysis of $f ' \left(x\right)$

$f ' \left(x\right) = 4 {x}^{3} - 12 {x}^{2}$ is never undefined and is $0$ at $x = 0$ and at $x = 3$. Investigating the sign of $f ' \left(x\right)$ we find

on $\left(- \infty , 0\right)$, $f ' \left(x\right)$ is negative, so $f$ is decreasing,
on $\left(0 , 3\right)$, $f ' \left(x\right)$ is negative, so $f$ is decreasing,
on $\left(3 , \infty\right)$, $f ' \left(x\right)$ is positive, so $f$ is increasing.

There is a relative minimum at $x = 3$. The minimum is $f \left(3\right) = 3 {\left(3\right)}^{3} - 4 {\left(3\right)}^{3} = - 27$

There are no relative maxima.

Before we look at concavity, here is the straight line sketch:

Analysis of $f ' ' \left(x\right)$

$f ' ' \left(x\right) = 12 {x}^{2} - 24 x = 12 x \left(x - 2\right)$ is never undefined and is $0$ at $x = 0$ and at $x = 2$. Investigating the sign of $f ' '$ we have

on $\left(- \infty , 0\right)$, $f ' ' \left(x\right)$ is positive, so $f$ is concave up (convex),
on $\left(0 , 3\right)$, $f ' ' \left(x\right)$ is negative, so $f$ is concave down (concave),
on $\left(3 , \infty\right)$, $f ' ' \left(x\right)$ is positive, so $f$ is concave up (convex).

Inflection points are points on the graph at which the concavity changed. Therefore, there are inflection points at $x = 0$ and $x = 2$.

The inflection points are: $\left(0 , 0\right)$ and $\left(2 , - 16\right)$.

(f(2) = 2(8)-4(8) = -16).

Now that we have concavity, we can improve our sketch:

Here is Socratic's graph. (You can move it and zoom in/out. If you leave this answer and come back, the graph will reset to the original view.)
graph{x^4-4x^3 [-37.2, 35.86, -29.07, 7.48]}