Question

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

Answer 1

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 - 5x = x(x^4 - 5)`

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 = +-root4(5)`
Now you have the x-intercepts. Put them on a graph.
Observe that `x = +-root4(5) > 1`.

Next step: extrema. Take the first derivative.
Letting y = f(x), we have
`f'(x) = 5x^4 - 5`
`= 5(x^4 - 1)`
`= 5(x^2 + 1)(x^2 - 1)`
`= 5(x^2 + 1)(x+1)(x-1)`
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''(x) = 20x^3`, you may use the Second Derivative test to determine whether there is a max or a min at each critical point.
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.