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

##### 1 Answer

First step: factoring!

#### Explanation:

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

To find the x-intercepts, factor the polynomial completely.

This gives us

The first equation is already solved. There is an x-intercept at (0, 0).

The second equation has two solutions.

Now you have the x-intercepts. Put them on a graph.

Observe that

Next step: extrema. Take the first derivative.

Letting y = f(x), we have

Now f' is defined for all x and is zero only at

These are your critical values (possible extrema).

Since

Find their y-coordinates, and add them to your graph.

Inflection points? Where is f'' equal to zero? Does f'' change sign at that location? If so, it's an inflection point.

A polynomial has no asymptotes.

You probably will not need to add more points in order to finish your graph, but perhaps plot the graph for x = -2 and x = 2.

These should complete it.