# How do you find intercepts, extrema, points of inflections, asymptotes and graph y=x^5-5x?

Jan 16, 2018

First step: factoring!

#### Explanation:

The y-intercept is (0, 0), since y(0) = 0.

To find the x-intercepts, factor the polynomial completely.
${x}^{5} - 5 x = x \left({x}^{4} - 5\right)$

This gives us $x = 0$ or ${x}^{4} - 5 = 0$.
The first equation is already solved. There is an x-intercept at (0, 0).
The second equation has two solutions.
${x}^{4} - 5 = 0$
${x}^{4} = 5$
$x = \pm \sqrt[4]{5}$
Now you have the x-intercepts. Put them on a graph.
Observe that $x = \pm \sqrt[4]{5} > 1$.

Next step: extrema. Take the first derivative.
Letting y = f(x), we have
$f ' \left(x\right) = 5 {x}^{4} - 5$
$= 5 \left({x}^{4} - 1\right)$
$= 5 \left({x}^{2} + 1\right) \left({x}^{2} - 1\right)$
$= 5 \left({x}^{2} + 1\right) \left(x + 1\right) \left(x - 1\right)$
Now f' is defined for all x and is zero only at $x = 1$ and $x = - 1$.

These are your critical values (possible extrema).
Since $f ' ' \left(x\right) = 20 {x}^{3}$, you may use the Second Derivative test to determine whether there is a max or a min at each critical point.